:: JORDAN7 semantic presentation

begin

theorem :: JORDAN7:1
for P being ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) st P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) is being_simple_closed_curve holds
( W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) in Lower_Arc P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( non empty ) ( non empty functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) & E-max P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) in Lower_Arc P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( non empty ) ( non empty functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) & W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) in Upper_Arc P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( non empty ) ( non empty functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) & E-max P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) in Upper_Arc P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( non empty ) ( non empty functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN7:2
for P being ( ( non empty compact ) ( non empty functional closed compact ) Subset of )
for q being ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) is being_simple_closed_curve & LE q : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) , W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) holds
q : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) = W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN7:3
for P being ( ( non empty compact ) ( non empty functional closed compact ) Subset of )
for q being ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) is being_simple_closed_curve & q : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) in P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) holds
LE W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,q : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ;

definition
let P be ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ;
let q1, q2 be ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ;
func Segment (q1,q2,P) -> ( ( ) ( functional ) Subset of ) equals :: JORDAN7:def 1
 { p : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) where p is ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) : ( LE q1 : ( ( ) ( ) set ) ,p : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( ) 2-sorted ) & LE p : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( ) Element of P : ( ( ) ( ) 2-sorted ) ) ,P : ( ( ) ( ) 2-sorted ) ) } if q2 : ( ( ) ( ) Element of P : ( ( ) ( ) 2-sorted ) ) <> W-min P : ( ( ) ( ) 2-sorted ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) )
otherwise { p1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) where p1 is ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) : ( LE q1 : ( ( ) ( ) set ) ,p1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( ) 2-sorted ) or ( q1 : ( ( ) ( ) set ) in P : ( ( ) ( ) 2-sorted ) & p1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) = W-min P : ( ( ) ( ) 2-sorted ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ) } ;
end;

theorem :: JORDAN7:4
for P being ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) st P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) is being_simple_closed_curve holds
( Segment ((W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(E-max P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( functional ) Subset of ) = Upper_Arc P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( non empty ) ( non empty functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) & Segment ((E-max P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( functional ) Subset of ) = Lower_Arc P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( non empty ) ( non empty functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN7:5
for P being ( ( non empty compact ) ( non empty functional closed compact ) Subset of )
for q1, q2 being ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) is being_simple_closed_curve & LE q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) holds
( q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) in P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) & q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) in P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) ;

theorem :: JORDAN7:6
for P being ( ( non empty compact ) ( non empty functional closed compact ) Subset of )
for q1, q2 being ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) is being_simple_closed_curve & LE q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) holds
( q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) in Segment (q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( functional ) Subset of ) & q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) in Segment (q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( functional ) Subset of ) ) ;

theorem :: JORDAN7:7
for P being ( ( non empty compact ) ( non empty functional closed compact ) Subset of )
for q1 being ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) st q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) in P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) & P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) is being_simple_closed_curve holds
q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) in Segment (q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,(W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( functional ) Subset of ) ;

theorem :: JORDAN7:8
for P being ( ( non empty compact ) ( non empty functional closed compact ) Subset of )
for q being ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) is being_simple_closed_curve & q : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) in P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) & q : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) <> W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) holds
Segment (q : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( functional ) Subset of ) = {q : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty functional ) set ) ;

theorem :: JORDAN7:9
for P being ( ( non empty compact ) ( non empty functional closed compact ) Subset of )
for q1, q2 being ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) is being_simple_closed_curve & q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) <> W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) & q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) <> W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) holds
not W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) in Segment (q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( functional ) Subset of ) ;

theorem :: JORDAN7:10
for P being ( ( non empty compact ) ( non empty functional closed compact ) Subset of )
for q1, q2, q3 being ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) is being_simple_closed_curve & LE q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) & LE q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q3 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) & ( not q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) = q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) or not q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) = W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) & ( not q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) = q3 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) or not q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) = W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) holds
(Segment (q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) )) : ( ( ) ( functional ) Subset of ) /\ (Segment (q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q3 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) )) : ( ( ) ( functional ) Subset of ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = {q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty functional ) set ) ;

theorem :: JORDAN7:11
for P being ( ( non empty compact ) ( non empty functional closed compact ) Subset of )
for q1, q2 being ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) is being_simple_closed_curve & LE q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) & q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) <> W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) & q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) <> W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) holds
(Segment (q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) )) : ( ( ) ( functional ) Subset of ) /\ (Segment (q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,(W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) )) : ( ( ) ( functional ) Subset of ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = {q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty functional ) set ) ;

theorem :: JORDAN7:12
for P being ( ( non empty compact ) ( non empty functional closed compact ) Subset of )
for q1, q2 being ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) is being_simple_closed_curve & LE q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) & q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) <> q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) & q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) <> W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) holds
(Segment (q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,(W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) )) : ( ( ) ( functional ) Subset of ) /\ (Segment ((W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) )) : ( ( ) ( functional ) Subset of ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = {(W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty functional ) set ) ;

theorem :: JORDAN7:13
for P being ( ( non empty compact ) ( non empty functional closed compact ) Subset of )
for q1, q2, q3, q4 being ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) is being_simple_closed_curve & LE q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) & LE q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q3 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) & LE q3 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q4 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) & q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) <> q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) & q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) <> q3 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) holds
Segment (q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( functional ) Subset of ) misses Segment (q3 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q4 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( functional ) Subset of ) ;

theorem :: JORDAN7:14
for P being ( ( non empty compact ) ( non empty functional closed compact ) Subset of )
for q1, q2, q3 being ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) is being_simple_closed_curve & LE q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) & LE q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q3 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) & q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) <> W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) & q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) <> q3 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) holds
Segment (q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( functional ) Subset of ) misses Segment (q3 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,(W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( functional ) Subset of ) ;

begin

theorem :: JORDAN7:15
for n being ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) )
for P being ( ( non empty ) ( non empty functional ) Subset of )
for f being ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of ((TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) | b2 : ( ( non empty ) ( non empty functional ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) ) : ( ( ) ( non empty ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of ((TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) | b2 : ( ( non empty ) ( non empty functional ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) ) : ( ( ) ( non empty ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of ((TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) | b2 : ( ( non empty ) ( non empty functional ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty ) set ) ) st f : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of ((TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) | b2 : ( ( non empty ) ( non empty functional ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) ) : ( ( ) ( non empty ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of ((TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) | b2 : ( ( non empty ) ( non empty functional ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) ) : ( ( ) ( non empty ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of ((TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) | b2 : ( ( non empty ) ( non empty functional ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty ) set ) ) is being_homeomorphism holds
ex g being ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) st
( f : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of ((TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) | b2 : ( ( non empty ) ( non empty functional ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) ) : ( ( ) ( non empty ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of ((TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) | b2 : ( ( non empty ) ( non empty functional ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) ) : ( ( ) ( non empty ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of ((TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) | b2 : ( ( non empty ) ( non empty functional ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty ) set ) ) = g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) & g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) is continuous & g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) is one-to-one ) ;

theorem :: JORDAN7:16
for n being ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) )
for P being ( ( non empty ) ( non empty functional ) Subset of )
for g being ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) st g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) is continuous & g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) is one-to-one & rng g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = P : ( ( non empty ) ( non empty functional ) Subset of ) holds
ex f being ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of ((TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) | b2 : ( ( non empty ) ( non empty functional ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) ) : ( ( ) ( non empty ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of ((TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) | b2 : ( ( non empty ) ( non empty functional ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) ) : ( ( ) ( non empty ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of ((TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) | b2 : ( ( non empty ) ( non empty functional ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty ) set ) ) st
( f : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of ((TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) | b2 : ( ( non empty ) ( non empty functional ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) ) : ( ( ) ( non empty ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of ((TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) | b2 : ( ( non empty ) ( non empty functional ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) ) : ( ( ) ( non empty ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of ((TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) | b2 : ( ( non empty ) ( non empty functional ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty ) set ) ) = g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) & f : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of ((TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) | b2 : ( ( non empty ) ( non empty functional ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) ) : ( ( ) ( non empty ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of ((TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) | b2 : ( ( non empty ) ( non empty functional ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) ) : ( ( ) ( non empty ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of ((TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) | b2 : ( ( non empty ) ( non empty functional ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL b1 : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty ) set ) ) is being_homeomorphism ) ;

theorem :: JORDAN7:17
for A being ( ( ) ( functional ) Subset of )
for p1, p2 being ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) st A : ( ( ) ( functional ) Subset of ) is_an_arc_of p1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) holds
ex g being ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) st
( g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) is continuous & g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) is one-to-one & rng g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = A : ( ( ) ( functional ) Subset of ) & g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) . 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative V19() Function-like functional FinSequence-membered V122() V169() V170() V171() V172() V173() V174() V175() V208() V211() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() Function-like ) set ) = p1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) & g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) . 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() Function-like ) set ) = p2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ) ;

theorem :: JORDAN7:18
for P being ( ( non empty ) ( non empty functional ) Subset of )
for p1, p2, q1, q2 being ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) )
for g being ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) )
for s1, s2 being ( ( ) ( V11() V12() ext-real ) Real) st P : ( ( non empty ) ( non empty functional ) Subset of ) is_an_arc_of p1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) & g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) is continuous & g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) is one-to-one & rng g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = P : ( ( non empty ) ( non empty functional ) Subset of ) & g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) . 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative V19() Function-like functional FinSequence-membered V122() V169() V170() V171() V172() V173() V174() V175() V208() V211() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() Function-like ) set ) = p1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) & g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) . 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() Function-like ) set ) = p2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) & g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) . s1 : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V21() Function-like ) set ) = q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) & 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative V19() Function-like functional FinSequence-membered V122() V169() V170() V171() V172() V173() V174() V175() V208() V211() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) <= s1 : ( ( ) ( V11() V12() ext-real ) Real) & s1 : ( ( ) ( V11() V12() ext-real ) Real) <= 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) & g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) . s2 : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V21() Function-like ) set ) = q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) & 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative V19() Function-like functional FinSequence-membered V122() V169() V170() V171() V172() V173() V174() V175() V208() V211() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) <= s2 : ( ( ) ( V11() V12() ext-real ) Real) & s2 : ( ( ) ( V11() V12() ext-real ) Real) <= 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) & s1 : ( ( ) ( V11() V12() ext-real ) Real) <= s2 : ( ( ) ( V11() V12() ext-real ) Real) holds
LE q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty ) ( non empty functional ) Subset of ) ,p1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN7:19
for P being ( ( non empty ) ( non empty functional ) Subset of )
for p1, p2, q1, q2 being ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) )
for g being ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) )
for s1, s2 being ( ( ) ( V11() V12() ext-real ) Real) st g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) is continuous & g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) is one-to-one & rng g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = P : ( ( non empty ) ( non empty functional ) Subset of ) & g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) . 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative V19() Function-like functional FinSequence-membered V122() V169() V170() V171() V172() V173() V174() V175() V208() V211() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() Function-like ) set ) = p1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) & g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) . 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() Function-like ) set ) = p2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) & g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) . s1 : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V21() Function-like ) set ) = q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) & 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative V19() Function-like functional FinSequence-membered V122() V169() V170() V171() V172() V173() V174() V175() V208() V211() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) <= s1 : ( ( ) ( V11() V12() ext-real ) Real) & s1 : ( ( ) ( V11() V12() ext-real ) Real) <= 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) & g : ( ( Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) ( V21() V24( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V35( the carrier of I[01] : ( ( ) ( TopSpace-like T_0 T_1 T_2 compact ) SubSpace of K293() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty functional ) set ) ) . s2 : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V21() Function-like ) set ) = q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) & 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative V19() Function-like functional FinSequence-membered V122() V169() V170() V171() V172() V173() V174() V175() V208() V211() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) <= s2 : ( ( ) ( V11() V12() ext-real ) Real) & s2 : ( ( ) ( V11() V12() ext-real ) Real) <= 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) & LE q1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty ) ( non empty functional ) Subset of ) ,p1 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Point of ( ( ) ( non empty functional ) set ) ) holds
s1 : ( ( ) ( V11() V12() ext-real ) Real) <= s2 : ( ( ) ( V11() V12() ext-real ) Real) ;

theorem :: JORDAN7:20
for P being ( ( non empty compact ) ( non empty functional closed compact ) Subset of )
for e being ( ( ) ( V11() V12() ext-real ) Real) st P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) is being_simple_closed_curve & e : ( ( ) ( V11() V12() ext-real ) Real) > 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative V19() Function-like functional FinSequence-membered V122() V169() V170() V171() V172() V173() V174() V175() V208() V211() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) holds
ex h being ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) st
( h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) . 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() Function-like ) set ) = W-min P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) & h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) is one-to-one & 8 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) <= len h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) & rng h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) c= P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) & ( for i being ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) < len h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) holds
LE h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. (i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) & ( for i being ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) )
for W being ( ( ) ( ) Subset of ) st 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) < len h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) & W : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) = Segment ((h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. (i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( functional ) Subset of ) holds
diameter W : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) < e : ( ( ) ( V11() V12() ext-real ) Real) ) & ( for W being ( ( ) ( ) Subset of ) st W : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) = Segment ((h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. (len h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( functional ) Subset of ) holds
diameter W : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) < e : ( ( ) ( V11() V12() ext-real ) Real) ) & ( for i being ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) < len h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(Segment ((h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. (i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) )) : ( ( ) ( functional ) Subset of ) /\ (Segment ((h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. (i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. (i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) )) : ( ( ) ( functional ) Subset of ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = {(h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. (i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty functional ) set ) ) & (Segment ((h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. (len h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) )) : ( ( ) ( functional ) Subset of ) /\ (Segment ((h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) )) : ( ( ) ( functional ) Subset of ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = {(h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty functional ) set ) & (Segment ((h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. ((len h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) -' 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V10() V11() V12() ext-real non negative V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. (len h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) )) : ( ( ) ( functional ) Subset of ) /\ (Segment ((h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. (len h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) )) : ( ( ) ( functional ) Subset of ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = {(h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. (len h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty functional ) set ) & Segment ((h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. ((len h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) -' 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V10() V11() V12() ext-real non negative V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. (len h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( functional ) Subset of ) misses Segment ((h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( functional ) Subset of ) & ( for i, j being ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) < j : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) < len h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) & not i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) are_adjacent1 holds
Segment ((h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. (i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( functional ) Subset of ) misses Segment ((h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. j : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. (j : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( functional ) Subset of ) ) & ( for i being ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) < i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) < len h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) holds
Segment ((h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. (len h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( functional ) Subset of ) misses Segment ((h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(h : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) Function-like V38() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) /. (i : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V10() V11() V12() ext-real V19() V122() V169() V170() V171() V172() V173() V174() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V38() V45(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like V159() V160() V161() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V19() V122() V169() V170() V171() V172() V173() V174() V206() V208() ) Element of NAT : ( ( ) ( V169() V170() V171() V172() V173() V174() V175() V208() ) Element of K6(REAL : ( ( ) ( V169() V170() V171() V175() V208() V209() V211() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V193() ) ( non empty TopSpace-like T_0 T_1 T_2 V135() V181() V182() V183() V184() V185() V186() V187() V193() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,P : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( functional ) Subset of ) ) ) ;