( 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 ) ) ) ,3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 ) ) ) ,3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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, z 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 ) ) ) ,3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 ) ) ,z : ( ( ) ( 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 complex-valued ext-real-valued real-valued V54() V61(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 ) ) ) ) ;

( 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 ) ) ) ,3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 ) ) ) ,3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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, z 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 ) ) ) ,3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 ) ) ,z : ( ( ) ( 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 complex-valued ext-real-valued real-valued V54() V61(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 ) ) ) ) ;

( dom (proj (3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 (3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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, z being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) holds (proj (3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 ) ) ,z : ( ( ) ( 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 complex-valued ext-real-valued real-valued V54() V61(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 ) ) = z : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ) ;

for x, y, z being ( ( ) ( V11() real ext-real ) Real)
for u 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(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 u : ( ( ) ( 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(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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) ,z : ( ( ) ( 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 complex-valued ext-real-valued real-valued V54() V61(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 u : ( ( ) ( 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(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 ,) ,u : ( ( ) ( 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(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 ) 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 ) ) is_differentiable_in x : ( ( ) ( V11() real ext-real ) Real) ;

for x, y, z being ( ( ) ( V11() real ext-real ) Real)
for u 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(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 u : ( ( ) ( 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(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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) ,z : ( ( ) ( 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 complex-valued ext-real-valued real-valued V54() V61(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 u : ( ( ) ( 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(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 ,) ,u : ( ( ) ( 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(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 ) 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 ) ) is_differentiable_in y : ( ( ) ( V11() real ext-real ) Real) ;

for x, y, z being ( ( ) ( V11() real ext-real ) Real)
for u 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(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 u : ( ( ) ( 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(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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) ,z : ( ( ) ( 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 complex-valued ext-real-valued real-valued V54() V61(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 u : ( ( ) ( 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(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 (3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 ,) ,u : ( ( ) ( 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(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 ) 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 ) ) is_differentiable_in z : ( ( ) ( V11() real ext-real ) Real) ;

for f being ( ( Function-like ) ( V16() V19( REAL 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 u 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(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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, z0 being ( ( ) ( V11() real ext-real ) Real) st
( u : ( ( ) ( 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(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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) ,z0 : ( ( ) ( 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 complex-valued ext-real-valued real-valued V54() V61(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 ,) ,u : ( ( ) ( 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(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 ) 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 ) ) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) ) iff f : ( ( Function-like ) ( V16() V19( REAL 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 u : ( ( ) ( 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(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 ) ) ) ) ;

for f being ( ( Function-like ) ( V16() V19( REAL 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 u 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(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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, z0 being ( ( ) ( V11() real ext-real ) Real) st
( u : ( ( ) ( 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(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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) ,z0 : ( ( ) ( 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 complex-valued ext-real-valued real-valued V54() V61(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 ,) ,u : ( ( ) ( 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(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 ) 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 ) ) is_differentiable_in y0 : ( ( ) ( V11() real ext-real ) Real) ) iff f : ( ( Function-like ) ( V16() V19( REAL 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 u : ( ( ) ( 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(3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() 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 3 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of 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 ) ) ) ) ;