:: NCFCONT1 semantic presentation

begin

theorem :: NCFCONT1:1
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) holds - seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (- 1r : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) * seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

definition
let CNS be ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) ;
let x0 be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
mode Neighbourhood of x0 -> ( ( ) ( ) Subset of ) means :: NCFCONT1:def 1
 ex g being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < g : ( ( ) ( V11() real ext-real ) Real) & { y : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) where y is ( ( ) ( ) Point of ( ( ) ( ) set ) ) : ||.(y : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x0 : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) ) : ( ( ) ( ) Element of the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < g : ( ( ) ( V11() real ext-real ) Real) } c= it : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) );
end;

theorem :: NCFCONT1:2
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for g being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < g : ( ( ) ( V11() real ext-real ) Real) holds
{ y : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) where y is ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ||.(y : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < g : ( ( ) ( V11() real ext-real ) Real) } is ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: NCFCONT1:3
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for N being ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) holds x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in N : ( ( ) ( ) Neighbourhood of b2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

definition
let CNS be ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) ;
let X be ( ( ) ( ) Subset of ) ;
attr X is compact means :: NCFCONT1:def 2
 for s1 being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( ) set ) ) st rng s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= X : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) holds
ex s2 being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( ) set ) ) st
( s2 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) subsequence of s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) & s2 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is convergent & lim s2 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) in X : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) );
end;

definition
let CNS be ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) ;
let X be ( ( ) ( ) Subset of ) ;
attr X is closed means :: NCFCONT1:def 3
 for s1 being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( ) set ) ) st rng s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= X : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) & s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is convergent holds
lim s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) in X : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) ;
end;

definition
let CNS be ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) ;
let X be ( ( ) ( ) Subset of ) ;
attr X is open means :: NCFCONT1:def 4
X : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) ` : ( ( ) ( ) Element of K6( the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is closed ;
end;

definition
let CNS1, CNS2 be ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) ;
let f be ( ( Function-like ) ( Relation-like the carrier of CNS1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of CNS2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ;
let x0 be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
pred f is_continuous_in x0 means :: NCFCONT1:def 5
( x0 : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS1 : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) in dom f : ( ( Function-like quasi_total ) ( Relation-like K7(CNS1 : ( ( ) ( ) NORMSTR ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS1 : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS1 : ( ( ) ( ) NORMSTR ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) Element of K6( the carrier of CNS1 : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ( for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS1 : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( ) set ) ) st rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of CNS1 : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( Function-like quasi_total ) ( Relation-like K7(CNS1 : ( ( ) ( ) NORMSTR ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS1 : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS1 : ( ( ) ( ) NORMSTR ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) Element of K6( the carrier of CNS1 : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is convergent & lim seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of CNS1 : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) = x0 : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS1 : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
( f : ( ( Function-like quasi_total ) ( Relation-like K7(CNS1 : ( ( ) ( ) NORMSTR ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS1 : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS1 : ( ( ) ( ) NORMSTR ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS2 : ( ( ) ( ) Element of CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of CNS2 : ( ( ) ( ) Element of CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is convergent & f : ( ( Function-like quasi_total ) ( Relation-like K7(CNS1 : ( ( ) ( ) NORMSTR ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS1 : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS1 : ( ( ) ( ) NORMSTR ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x0 : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS1 : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of CNS2 : ( ( ) ( ) Element of CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) = lim (f : ( ( Function-like quasi_total ) ( Relation-like K7(CNS1 : ( ( ) ( ) NORMSTR ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS1 : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS1 : ( ( ) ( ) NORMSTR ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS2 : ( ( ) ( ) Element of CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of CNS2 : ( ( ) ( ) Element of CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of CNS2 : ( ( ) ( ) Element of CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) ) ) );
end;

definition
let CNS be ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) ;
let RNS be ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) ;
let f be ( ( Function-like ) ( Relation-like the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of RNS : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ;
let x0 be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
pred f is_continuous_in x0 means :: NCFCONT1:def 6
( x0 : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) in dom f : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) Element of K6( the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ( for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( ) set ) ) st rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) Element of K6( the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is convergent & lim seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) = x0 : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
( f : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of RNS : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of RNS : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is convergent & f : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x0 : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of RNS : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) = lim (f : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of RNS : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of RNS : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of RNS : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) ) ) );
end;

definition
let RNS be ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) ;
let CNS be ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) ;
let f be ( ( Function-like ) ( Relation-like the carrier of RNS : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ;
let x0 be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
pred f is_continuous_in x0 means :: NCFCONT1:def 7
( x0 : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) in dom f : ( ( Function-like quasi_total ) ( Relation-like K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) Element of K6( the carrier of RNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ( for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of RNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( ) set ) ) st rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of RNS : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of RNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( Function-like quasi_total ) ( Relation-like K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) Element of K6( the carrier of RNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of RNS : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is convergent & lim seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of RNS : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of RNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) = x0 : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
( f : ( ( Function-like quasi_total ) ( Relation-like K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of RNS : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( ) ( ) Element of RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of CNS : ( ( ) ( ) Element of RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is convergent & f : ( ( Function-like quasi_total ) ( Relation-like K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x0 : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of CNS : ( ( ) ( ) Element of RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) = lim (f : ( ( Function-like quasi_total ) ( Relation-like K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of RNS : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( ) ( ) Element of RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of CNS : ( ( ) ( ) Element of RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of CNS : ( ( ) ( ) Element of RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) ) ) );
end;

definition
let CNS be ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) ;
let f be ( ( Function-like ) ( Relation-like the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) ;
let x0 be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
pred f is_continuous_in x0 means :: NCFCONT1:def 8
( x0 : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) in dom f : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) Element of K6( the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ( for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( ) set ) ) st rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) Element of K6( the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is convergent & lim seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) = x0 : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
( f : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like total quasi_total complex-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) is convergent & f : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) /. x0 : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) = lim (f : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like total quasi_total complex-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) ) ) );
end;

definition
let CNS be ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) ;
let f be ( ( Function-like ) ( Relation-like the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let x0 be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
pred f is_continuous_in x0 means :: NCFCONT1:def 9
( x0 : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) in dom f : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) Element of K6( the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ( for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( ) set ) ) st rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) Element of K6( the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is convergent & lim seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) = x0 : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
( f : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is convergent & f : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) /. x0 : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) = lim (f : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ) );
end;

definition
let RNS be ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) ;
let f be ( ( Function-like ) ( Relation-like the carrier of RNS : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) ;
let x0 be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
pred f is_continuous_in x0 means :: NCFCONT1:def 10
( x0 : ( ( Function-like quasi_total ) ( Relation-like K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) in dom f : ( ( ) ( ) Element of RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) Element of K6( the carrier of RNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ( for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of RNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( ) set ) ) st rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of RNS : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of RNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( ) ( ) Element of RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) Element of K6( the carrier of RNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of RNS : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is convergent & lim seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of RNS : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of RNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) = x0 : ( ( Function-like quasi_total ) ( Relation-like K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
( f : ( ( ) ( ) Element of RNS : ( ( ) ( ) NORMSTR ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of RNS : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like total quasi_total complex-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) is convergent & f : ( ( ) ( ) Element of RNS : ( ( ) ( ) NORMSTR ) ) /. x0 : ( ( Function-like quasi_total ) ( Relation-like K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) = lim (f : ( ( ) ( ) Element of RNS : ( ( ) ( ) NORMSTR ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of RNS : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like total quasi_total complex-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) ) ) );
end;

theorem :: NCFCONT1:4
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) )
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) )
for h being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= dom h : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) in dom h : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:5
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) )
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) )
for h being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= dom h : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) in dom h : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:6
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) )
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) )
for h being ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= dom h : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) in dom h : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:7
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) )
for x being ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) in rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) iff ex n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) st x : ( ( ) ( ) set ) = seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: NCFCONT1:8
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff ( x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for r being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
||.((f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) - (f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) ) ) ) ) ) ;

theorem :: NCFCONT1:9
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff ( x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for r being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
||.((f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) - (f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) ) ) ) ) ) ;

theorem :: NCFCONT1:10
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff ( x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for r being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
||.((f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) - (f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) ) ) ) ) ) ;

theorem :: NCFCONT1:11
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff ( x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for r being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
abs ((f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) - (f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) ) ) ) ) ) ;

theorem :: NCFCONT1:12
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff ( x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for r being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
|.((f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) - (f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) .| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) ) ) ) ) ) ;

theorem :: NCFCONT1:13
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff ( x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for r being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
|.((f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) - (f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) .| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) ) ) ) ) ) ;

theorem :: NCFCONT1:14
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff ( x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for N1 being ( ( ) ( ) Neighbourhood of f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ex N being ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) st
for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in N : ( ( ) ( ) Neighbourhood of b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) in N1 : ( ( ) ( ) Neighbourhood of b3 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ) ) ) ;

theorem :: NCFCONT1:15
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff ( x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for N1 being ( ( ) ( ) Neighbourhood of f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ex N being ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) st
for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in N : ( ( ) ( ) Neighbourhood of b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) in N1 : ( ( ) ( ) Neighbourhood of b3 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ) ) ) ;

theorem :: NCFCONT1:16
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff ( x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for N1 being ( ( ) ( ) Neighbourhood of f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ex N being ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) st
for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in N : ( ( ) ( ) Neighbourhood of b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) in N1 : ( ( ) ( ) Neighbourhood of b3 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ) ) ) ;

theorem :: NCFCONT1:17
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff ( x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for N1 being ( ( ) ( ) Neighbourhood of f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ex N being ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .: N : ( ( ) ( ) Neighbourhood of b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= N1 : ( ( ) ( ) Neighbourhood of b3 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ) ) ) ;

theorem :: NCFCONT1:18
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff ( x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for N1 being ( ( ) ( ) Neighbourhood of f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ex N being ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .: N : ( ( ) ( ) Neighbourhood of b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= N1 : ( ( ) ( ) Neighbourhood of b3 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ) ) ) ;

theorem :: NCFCONT1:19
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff ( x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for N1 being ( ( ) ( ) Neighbourhood of f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ex N being ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .: N : ( ( ) ( ) Neighbourhood of b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= N1 : ( ( ) ( ) Neighbourhood of b3 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ) ) ) ;

theorem :: NCFCONT1:20
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ex N being ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) st (dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ N : ( ( ) ( ) Neighbourhood of b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: NCFCONT1:21
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ex N being ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) st (dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ N : ( ( ) ( ) Neighbourhood of b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: NCFCONT1:22
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ex N being ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) st (dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ N : ( ( ) ( ) Neighbourhood of b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: NCFCONT1:23
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for h1, h2 being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) st rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= (dom h1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (dom h2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
( (h1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) + h2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (h1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) + (h2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & (h1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) - h2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (h1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (h2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: NCFCONT1:24
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for h1, h2 being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) st rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= (dom h1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (dom h2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
( (h1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) + h2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (h1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) + (h2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & (h1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) - h2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (h1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (h2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: NCFCONT1:25
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for h1, h2 being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) st rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= (dom h1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (dom h2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
( (h1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) + h2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (h1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) + (h2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & (h1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) - h2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (h1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (h2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: NCFCONT1:26
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for h being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) )
for z being ( ( V11() ) ( V11() ) Complex) st rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= dom h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
(z : ( ( V11() ) ( V11() ) Complex) (#) h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = z : ( ( V11() ) ( V11() ) Complex) * (h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:27
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for h being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) )
for r being ( ( ) ( V11() real ext-real ) Real) st rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= dom h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
(r : ( ( ) ( V11() real ext-real ) Real) (#) h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) * (h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:28
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for h being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) )
for z being ( ( V11() ) ( V11() ) Complex) st rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= dom h : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
(z : ( ( V11() ) ( V11() ) Complex) (#) h : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = z : ( ( V11() ) ( V11() ) Complex) * (h : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:29
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for h being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) st rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= dom h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
( ||.(h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = ||.h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) & - (h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (- h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: NCFCONT1:30
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for h being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) st rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= dom h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
( ||.(h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = ||.h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) & - (h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (- h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: NCFCONT1:31
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for h being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) st rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= dom h : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
( ||.(h : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = ||.h : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) & - (h : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (- h : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: NCFCONT1:32
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: NCFCONT1:33
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: NCFCONT1:34
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & f2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & f1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: NCFCONT1:35
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for z being ( ( V11() ) ( V11() ) Complex) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
z : ( ( V11() ) ( V11() ) Complex) (#) f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: NCFCONT1:36
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for r being ( ( ) ( V11() real ext-real ) Real) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: NCFCONT1:37
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for z being ( ( V11() ) ( V11() ) Complex) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
z : ( ( V11() ) ( V11() ) Complex) (#) f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: NCFCONT1:38
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( ||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & - f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: NCFCONT1:39
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( ||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & - f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: NCFCONT1:40
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( ||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & - f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

definition
let CNS1, CNS2 be ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) ;
let f be ( ( Function-like ) ( Relation-like the carrier of CNS1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of CNS2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ;
let X be ( ( ) ( ) set ) ;
pred f is_continuous_on X means :: NCFCONT1:def 11
( X : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS1 : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( Function-like quasi_total ) ( Relation-like K7(CNS1 : ( ( ) ( ) NORMSTR ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS1 : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS1 : ( ( ) ( ) NORMSTR ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) Element of K6( the carrier of CNS1 : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( ) Point of ( ( ) ( ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS1 : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like quasi_total ) ( Relation-like K7(CNS1 : ( ( ) ( ) NORMSTR ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS1 : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS1 : ( ( ) ( ) NORMSTR ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) | X : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS1 : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of CNS1 : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -defined the carrier of CNS2 : ( ( ) ( ) Element of CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -valued Function-like ) Element of K6(K7( the carrier of CNS1 : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) , the carrier of CNS2 : ( ( ) ( ) Element of CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) );
end;

definition
let CNS be ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) ;
let RNS be ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) ;
let f be ( ( Function-like ) ( Relation-like the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of RNS : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ;
let X be ( ( ) ( ) set ) ;
pred f is_continuous_on X means :: NCFCONT1:def 12
( X : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) Element of K6( the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( ) Point of ( ( ) ( ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) | X : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -defined the carrier of RNS : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -valued Function-like ) Element of K6(K7( the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) , the carrier of RNS : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) );
end;

definition
let RNS be ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) ;
let CNS be ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) ;
let g be ( ( Function-like ) ( Relation-like the carrier of RNS : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ;
let X be ( ( ) ( ) set ) ;
pred g is_continuous_on X means :: NCFCONT1:def 13
( X : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= dom g : ( ( Function-like quasi_total ) ( Relation-like K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) Element of K6( the carrier of RNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( ) Point of ( ( ) ( ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
g : ( ( Function-like quasi_total ) ( Relation-like K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) | X : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of RNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -defined the carrier of CNS : ( ( ) ( ) Element of RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -valued Function-like ) Element of K6(K7( the carrier of RNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) , the carrier of CNS : ( ( ) ( ) Element of RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) );
end;

definition
let CNS be ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) ;
let f be ( ( Function-like ) ( Relation-like the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) ;
let X be ( ( ) ( ) set ) ;
pred f is_continuous_on X means :: NCFCONT1:def 14
( X : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) Element of K6( the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( ) Point of ( ( ) ( ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) | X : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) Element of K6(K7( the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ,COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) );
end;

definition
let CNS be ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) ;
let f be ( ( Function-like ) ( Relation-like the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let X be ( ( ) ( ) set ) ;
pred f is_continuous_on X means :: NCFCONT1:def 15
( X : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) Element of K6( the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( ) Point of ( ( ) ( ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) | X : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) );
end;

definition
let RNS be ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) ;
let f be ( ( Function-like ) ( Relation-like the carrier of RNS : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) ;
let X be ( ( ) ( ) set ) ;
pred f is_continuous_on X means :: NCFCONT1:def 16
( X : ( ( Function-like quasi_total ) ( Relation-like K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( ) ( ) Element of RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) Element of K6( the carrier of RNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( ) Point of ( ( ) ( ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( Function-like quasi_total ) ( Relation-like K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( ) ( ) Element of RNS : ( ( ) ( ) NORMSTR ) ) | X : ( ( Function-like quasi_total ) ( Relation-like K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of RNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) Element of K6(K7( the carrier of RNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ,COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) );
end;

theorem :: NCFCONT1:41
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) iff ( X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for s1 being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) st rng s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= X : ( ( ) ( ) set ) & s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is convergent & lim s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is convergent & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. (lim s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) = lim (f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ) ) ) ;

theorem :: NCFCONT1:42
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) iff ( X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for s1 being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) st rng s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= X : ( ( ) ( ) set ) & s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is convergent & lim s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is convergent & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. (lim s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) = lim (f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ) ) ) ;

theorem :: NCFCONT1:43
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) iff ( X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for s1 being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) st rng s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= X : ( ( ) ( ) set ) & s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is convergent & lim s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is convergent & f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. (lim s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) = lim (f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ) ) ) ;

theorem :: NCFCONT1:44
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) iff ( X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for r being ( ( ) ( V11() real ext-real ) Real) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) & 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) & ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
||.((f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) - (f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) ) ) ) ) ) ;

theorem :: NCFCONT1:45
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) iff ( X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for r being ( ( ) ( V11() real ext-real ) Real) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) & 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) & ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
||.((f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) - (f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) ) ) ) ) ) ;

theorem :: NCFCONT1:46
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) iff ( X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for r being ( ( ) ( V11() real ext-real ) Real) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) & 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) & ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
||.((f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) - (f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) ) ) ) ) ) ;

theorem :: NCFCONT1:47
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) iff ( X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for r being ( ( ) ( V11() real ext-real ) Real) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) & 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) & ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
|.((f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) - (f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) .| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) ) ) ) ) ) ;

theorem :: NCFCONT1:48
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) iff ( X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for r being ( ( ) ( V11() real ext-real ) Real) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) & 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) & ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
abs ((f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) - (f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) ) ) ) ) ) ;

theorem :: NCFCONT1:49
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) iff ( X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for r being ( ( ) ( V11() real ext-real ) Real) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) & 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) & ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
|.((f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) - (f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) .| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) ) ) ) ) ) ;

theorem :: NCFCONT1:50
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) iff f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) | X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:51
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) iff f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) | X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:52
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) iff f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) | X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:53
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) iff f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) | X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:54
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) iff f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) | X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:55
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) iff f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) | X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:56
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X, X1 being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) & X1 : ( ( ) ( ) set ) c= X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X1 : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:57
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X, X1 being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) & X1 : ( ( ) ( ) set ) c= X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X1 : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:58
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X, X1 being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) & X1 : ( ( ) ( ) set ) c= X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X1 : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:59
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on {x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) ;

theorem :: NCFCONT1:60
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on {x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) ;

theorem :: NCFCONT1:61
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on {x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) ;

theorem :: NCFCONT1:62
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X being ( ( ) ( ) set )
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) & f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) holds
( f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) & f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:63
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) & f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) holds
( f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) & f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:64
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) & f2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) holds
( f1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) & f1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:65
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X, X1 being ( ( ) ( ) set )
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) & f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X1 : ( ( ) ( ) set ) holds
( f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) /\ X1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) & f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) /\ X1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:66
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X, X1 being ( ( ) ( ) set )
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) & f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X1 : ( ( ) ( ) set ) holds
( f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) /\ X1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) & f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) /\ X1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:67
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X, X1 being ( ( ) ( ) set )
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) & f2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X1 : ( ( ) ( ) set ) holds
( f1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) /\ X1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) & f1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) /\ X1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:68
for z being ( ( V11() ) ( V11() ) Complex)
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) holds
z : ( ( V11() ) ( V11() ) Complex) (#) f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:69
for r being ( ( ) ( V11() real ext-real ) Real)
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) holds
r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:70
for z being ( ( V11() ) ( V11() ) Complex)
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) holds
z : ( ( V11() ) ( V11() ) Complex) (#) f : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:71
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) holds
( ||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) & - f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:72
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) holds
( ||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) & - f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:73
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) holds
( ||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) & - f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:74
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is total & ( for x1, x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. (x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) + x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) = (f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) + (f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) & ex x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on the carrier of CNS1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ;

theorem :: NCFCONT1:75
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is total & ( for x1, x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. (x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) + x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) = (f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) + (f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) & ex x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ;

theorem :: NCFCONT1:76
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is total & ( for x1, x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. (x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) + x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) = (f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) + (f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) & ex x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on the carrier of RNS : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ;

theorem :: NCFCONT1:77
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is compact ;

theorem :: NCFCONT1:78
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is compact ;

theorem :: NCFCONT1:79
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is compact ;

theorem :: NCFCONT1:80
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) st dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) : ( ( ) ( V45() ) Element of K6(COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) is compact ;

theorem :: NCFCONT1:81
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( V45() V46() V47() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) is compact ;

theorem :: NCFCONT1:82
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) st dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) : ( ( ) ( V45() ) Element of K6(COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) is compact ;

theorem :: NCFCONT1:83
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for Y being ( ( ) ( ) Subset of )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st Y : ( ( ) ( ) Subset of ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Y : ( ( ) ( ) Subset of ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on Y : ( ( ) ( ) Subset of ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .: Y : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is compact ;

theorem :: NCFCONT1:84
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for Y being ( ( ) ( ) Subset of )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st Y : ( ( ) ( ) Subset of ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Y : ( ( ) ( ) Subset of ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on Y : ( ( ) ( ) Subset of ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .: Y : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is compact ;

theorem :: NCFCONT1:85
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for Y being ( ( ) ( ) Subset of )
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st Y : ( ( ) ( ) Subset of ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Y : ( ( ) ( ) Subset of ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on Y : ( ( ) ( ) Subset of ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .: Y : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is compact ;

theorem :: NCFCONT1:86
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) <> {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V45() V46() V47() V48() V49() V50() V51() ) set ) & dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
ex x1, x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st
( x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) = upper_bound (rng f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( V45() V46() V47() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) = lower_bound (rng f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( V45() V46() V47() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ;

theorem :: NCFCONT1:87
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) <> {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V45() V46() V47() V48() V49() V50() V51() ) set ) & dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
ex x1, x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st
( x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) = upper_bound (rng ||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V45() V46() V47() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) & ||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) = lower_bound (rng ||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V45() V46() V47() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ;

theorem :: NCFCONT1:88
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) <> {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V45() V46() V47() V48() V49() V50() V51() ) set ) & dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
ex x1, x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st
( x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) = upper_bound (rng ||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V45() V46() V47() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) & ||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) = lower_bound (rng ||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V45() V46() V47() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ;

theorem :: NCFCONT1:89
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) <> {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V45() V46() V47() V48() V49() V50() V51() ) set ) & dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
ex x1, x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st
( x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) = upper_bound (rng ||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V45() V46() V47() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) & ||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) = lower_bound (rng ||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V45() V46() V47() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ;

theorem :: NCFCONT1:90
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) holds ||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) | X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = ||.(f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:91
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) holds ||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) | X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = ||.(f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:92
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) holds ||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) | X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = ||.(f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:93
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for Y being ( ( ) ( ) Subset of ) st Y : ( ( ) ( ) Subset of ) <> {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V45() V46() V47() V48() V49() V50() V51() ) set ) & Y : ( ( ) ( ) Subset of ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Y : ( ( ) ( ) Subset of ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on Y : ( ( ) ( ) Subset of ) holds
ex x1, x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st
( x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in Y : ( ( ) ( ) Subset of ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in Y : ( ( ) ( ) Subset of ) & ||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) = upper_bound (||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) .: Y : ( ( ) ( ) Subset of ) ) : ( ( ) ( V45() V46() V47() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) & ||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) = lower_bound (||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) .: Y : ( ( ) ( ) Subset of ) ) : ( ( ) ( V45() V46() V47() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ;

theorem :: NCFCONT1:94
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for Y being ( ( ) ( ) Subset of ) st Y : ( ( ) ( ) Subset of ) <> {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V45() V46() V47() V48() V49() V50() V51() ) set ) & Y : ( ( ) ( ) Subset of ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Y : ( ( ) ( ) Subset of ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on Y : ( ( ) ( ) Subset of ) holds
ex x1, x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st
( x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in Y : ( ( ) ( ) Subset of ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in Y : ( ( ) ( ) Subset of ) & ||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) = upper_bound (||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) .: Y : ( ( ) ( ) Subset of ) ) : ( ( ) ( V45() V46() V47() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) & ||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) = lower_bound (||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) .: Y : ( ( ) ( ) Subset of ) ) : ( ( ) ( V45() V46() V47() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ;

theorem :: NCFCONT1:95
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for Y being ( ( ) ( ) Subset of ) st Y : ( ( ) ( ) Subset of ) <> {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V45() V46() V47() V48() V49() V50() V51() ) set ) & Y : ( ( ) ( ) Subset of ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Y : ( ( ) ( ) Subset of ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on Y : ( ( ) ( ) Subset of ) holds
ex x1, x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st
( x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in Y : ( ( ) ( ) Subset of ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in Y : ( ( ) ( ) Subset of ) & ||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) = upper_bound (||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) .: Y : ( ( ) ( ) Subset of ) ) : ( ( ) ( V45() V46() V47() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) & ||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) = lower_bound (||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) .: Y : ( ( ) ( ) Subset of ) ) : ( ( ) ( V45() V46() V47() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ;

theorem :: NCFCONT1:96
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for Y being ( ( ) ( ) Subset of ) st Y : ( ( ) ( ) Subset of ) <> {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V45() V46() V47() V48() V49() V50() V51() ) set ) & Y : ( ( ) ( ) Subset of ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Y : ( ( ) ( ) Subset of ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_continuous_on Y : ( ( ) ( ) Subset of ) holds
ex x1, x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st
( x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in Y : ( ( ) ( ) Subset of ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in Y : ( ( ) ( ) Subset of ) & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) = upper_bound (f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) .: Y : ( ( ) ( ) Subset of ) ) : ( ( ) ( V45() V46() V47() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) = lower_bound (f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) .: Y : ( ( ) ( ) Subset of ) ) : ( ( ) ( V45() V46() V47() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ;

definition
let CNS1, CNS2 be ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) ;
let X be ( ( ) ( ) set ) ;
let f be ( ( Function-like ) ( Relation-like the carrier of CNS1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of CNS2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ;
pred f is_Lipschitzian_on X means :: NCFCONT1:def 17
( X : ( ( Function-like quasi_total ) ( Relation-like K7(CNS1 : ( ( ) ( ) NORMSTR ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS1 : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS1 : ( ( ) ( ) NORMSTR ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS1 : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) Element of K6( the carrier of CNS1 : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & ( for x1, x2 being ( ( ) ( ) Point of ( ( ) ( ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( Function-like quasi_total ) ( Relation-like K7(CNS1 : ( ( ) ( ) NORMSTR ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS1 : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS1 : ( ( ) ( ) NORMSTR ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( Function-like quasi_total ) ( Relation-like K7(CNS1 : ( ( ) ( ) NORMSTR ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS1 : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS1 : ( ( ) ( ) NORMSTR ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
||.((f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS1 : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of CNS2 : ( ( ) ( ) Element of CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) - (f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS1 : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of CNS2 : ( ( ) ( ) Element of CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of CNS2 : ( ( ) ( ) Element of CNS1 : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) * ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of CNS1 : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ) );
end;

definition
let CNS be ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) ;
let RNS be ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) ;
let X be ( ( ) ( ) set ) ;
let f be ( ( Function-like ) ( Relation-like the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of RNS : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ;
pred f is_Lipschitzian_on X means :: NCFCONT1:def 18
( X : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) Element of K6( the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & ( for x1, x2 being ( ( ) ( ) Point of ( ( ) ( ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
||.((f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of RNS : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) - (f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of RNS : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of RNS : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) * ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ) );
end;

definition
let RNS be ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) ;
let CNS be ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) ;
let X be ( ( ) ( ) set ) ;
let f be ( ( Function-like ) ( Relation-like the carrier of RNS : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ;
pred f is_Lipschitzian_on X means :: NCFCONT1:def 19
( X : ( ( Function-like quasi_total ) ( Relation-like K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) Element of K6( the carrier of RNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & ( for x1, x2 being ( ( ) ( ) Point of ( ( ) ( ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( Function-like quasi_total ) ( Relation-like K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( Function-like quasi_total ) ( Relation-like K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
||.((f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of CNS : ( ( ) ( ) Element of RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) - (f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of CNS : ( ( ) ( ) Element of RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of CNS : ( ( ) ( ) Element of RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) * ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of RNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ) );
end;

definition
let CNS be ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) ;
let X be ( ( ) ( ) set ) ;
let f be ( ( Function-like ) ( Relation-like the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) ;
pred f is_Lipschitzian_on X means :: NCFCONT1:def 20
( X : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) c= dom f : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) Element of K6( the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & ( for x1, x2 being ( ( ) ( ) Point of ( ( ) ( ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) holds
|.((f : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) - (f : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) .| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) * ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ) );
end;

definition
let CNS be ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) ;
let X be ( ( ) ( ) set ) ;
let f be ( ( Function-like ) ( Relation-like the carrier of CNS : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
pred f is_Lipschitzian_on X means :: NCFCONT1:def 21
( X : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) c= dom f : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) Element of K6( the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & ( for x1, x2 being ( ( ) ( ) Point of ( ( ) ( ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) Element of CNS : ( ( ) ( ) NORMSTR ) ) holds
abs ((f : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) - (f : ( ( Function-like quasi_total ) ( Relation-like K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined CNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(CNS : ( ( ) ( ) NORMSTR ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,CNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) * ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of CNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ) );
end;

definition
let RNS be ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) ;
let X be ( ( ) ( ) set ) ;
let f be ( ( Function-like ) ( Relation-like the carrier of RNS : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) ;
pred f is_Lipschitzian_on X means :: NCFCONT1:def 22
( X : ( ( ) ( ) Element of RNS : ( ( ) ( ) NORMSTR ) ) c= dom f : ( ( Function-like quasi_total ) ( Relation-like K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) Element of K6( the carrier of RNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative V43() V44() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K6(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & ( for x1, x2 being ( ( ) ( ) Point of ( ( ) ( ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) Element of RNS : ( ( ) ( ) NORMSTR ) ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) Element of RNS : ( ( ) ( ) NORMSTR ) ) holds
|.((f : ( ( Function-like quasi_total ) ( Relation-like K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) - (f : ( ( Function-like quasi_total ) ( Relation-like K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined RNS : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) NORMSTR ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) ) .| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) * ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of RNS : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ) );
end;

theorem :: NCFCONT1:97
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X, X1 being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) & X1 : ( ( ) ( ) set ) c= X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X1 : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:98
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X, X1 being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) & X1 : ( ( ) ( ) set ) c= X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X1 : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:99
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X, X1 being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) & X1 : ( ( ) ( ) set ) c= X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X1 : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:100
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X, X1 being ( ( ) ( ) set )
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) & f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X1 : ( ( ) ( ) set ) holds
f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_Lipschitzian_on X : ( ( ) ( ) set ) /\ X1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:101
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X, X1 being ( ( ) ( ) set )
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) & f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X1 : ( ( ) ( ) set ) holds
f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_Lipschitzian_on X : ( ( ) ( ) set ) /\ X1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:102
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X, X1 being ( ( ) ( ) set )
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) & f2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X1 : ( ( ) ( ) set ) holds
f1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_Lipschitzian_on X : ( ( ) ( ) set ) /\ X1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:103
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X, X1 being ( ( ) ( ) set )
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) & f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X1 : ( ( ) ( ) set ) holds
f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_Lipschitzian_on X : ( ( ) ( ) set ) /\ X1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:104
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X, X1 being ( ( ) ( ) set )
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) & f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X1 : ( ( ) ( ) set ) holds
f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_Lipschitzian_on X : ( ( ) ( ) set ) /\ X1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:105
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X, X1 being ( ( ) ( ) set )
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) & f2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X1 : ( ( ) ( ) set ) holds
f1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_Lipschitzian_on X : ( ( ) ( ) set ) /\ X1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:106
for z being ( ( V11() ) ( V11() ) Complex)
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) holds
z : ( ( V11() ) ( V11() ) Complex) (#) f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_Lipschitzian_on X : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:107
for r being ( ( ) ( V11() real ext-real ) Real)
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) holds
r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_Lipschitzian_on X : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:108
for z being ( ( V11() ) ( V11() ) Complex)
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) holds
z : ( ( V11() ) ( V11() ) Complex) (#) f : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b3 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_Lipschitzian_on X : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:109
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) holds
( - f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_Lipschitzian_on X : ( ( ) ( ) set ) & ||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_Lipschitzian_on X : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:110
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) holds
( - f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_Lipschitzian_on X : ( ( ) ( ) set ) & ||.f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_Lipschitzian_on X : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:111
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) holds
( - f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_Lipschitzian_on X : ( ( ) ( ) set ) & ||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_Lipschitzian_on X : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:112
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) | X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is V20() holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:113
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) | X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is V20() holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:114
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) | X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is V20() holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:115
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for Y being ( ( ) ( ) Subset of ) holds id Y : ( ( ) ( ) Subset of ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_Lipschitzian_on Y : ( ( ) ( ) Subset of ) ;

theorem :: NCFCONT1:116
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:117
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:118
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:119
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:120
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:121
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V51() V52() ) set ) -valued Function-like complex-valued ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:122
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st ex r being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st rng f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {r : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:123
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st ex r being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st rng f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {r : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:124
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st ex r being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st rng f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {r : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:125
for CNS1, CNS2 being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) | X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is V20() holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:126
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) | X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is V20() holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:127
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for RNS being ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) | X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is V20() holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) ( non empty V116() V141() V142() V143() V144() V145() V146() V147() V151() V152() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:128
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st ( for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) = x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:129
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) = id (dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:130
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for Y being ( ( ) ( ) Subset of ) st Y : ( ( ) ( ) Subset of ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) | Y : ( ( ) ( ) Subset of ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = id Y : ( ( ) ( ) Subset of ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on Y : ( ( ) ( ) Subset of ) ;

theorem :: NCFCONT1:131
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for z being ( ( V11() ) ( V11() ) Complex)
for p being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) = (z : ( ( V11() ) ( V11() ) Complex) * x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) + p : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) ;

theorem :: NCFCONT1:132
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st ( for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) = ||.x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NCFCONT1:133
for CNS being ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) = ||.x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ( non empty V116() V141() V142() V143() V151() V152() V160() V161() V162() V163() ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) ;