:: JORDAN6 semantic presentation

begin

theorem :: JORDAN6:1
for r, s being ( ( real ) ( V11() real ext-real ) number ) st r : ( ( real ) ( V11() real ext-real ) number ) <= s : ( ( real ) ( V11() real ext-real ) number ) holds
( r : ( ( real ) ( V11() real ext-real ) number ) <= (r : ( ( real ) ( V11() real ext-real ) number ) + s : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) & (r : ( ( real ) ( V11() real ext-real ) number ) + s : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) <= s : ( ( real ) ( V11() real ext-real ) number ) ) ;

theorem :: JORDAN6:2
for TX being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for P being ( ( ) ( ) Subset of )
for A being ( ( ) ( ) Subset of )
for B being ( ( ) ( ) Subset of ) st B : ( ( ) ( ) Subset of ) is closed & A : ( ( ) ( ) Subset of ) = B : ( ( ) ( ) Subset of ) /\ P : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
A : ( ( ) ( ) Subset of ) is closed ;

theorem :: JORDAN6:3
for TX, TY being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for P being ( ( non empty ) ( non empty ) Subset of )
for f being ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V26( the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b3 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V26( the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b3 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V26( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & ( for f2 being ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V26( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st f2 : ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V26( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f : ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V26( the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b3 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V26( the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b3 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous holds
f2 : ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V26( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous ) ) ;

theorem :: JORDAN6:4
for r being ( ( real ) ( V11() real ext-real ) number )
for P being ( ( ) ( ) Subset of ) st P : ( ( ) ( ) Subset of ) = { p : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) where p is ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) : p : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) >= r : ( ( real ) ( V11() real ext-real ) number ) } holds
P : ( ( ) ( ) Subset of ) is closed ;

theorem :: JORDAN6:5
for r being ( ( real ) ( V11() real ext-real ) number )
for P being ( ( ) ( ) Subset of ) st P : ( ( ) ( ) Subset of ) = { p : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) where p is ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) : p : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) <= r : ( ( real ) ( V11() real ext-real ) number ) } holds
P : ( ( ) ( ) Subset of ) is closed ;

theorem :: JORDAN6:6
for r being ( ( real ) ( V11() real ext-real ) number )
for P being ( ( ) ( ) Subset of ) st P : ( ( ) ( ) Subset of ) = { p : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) where p is ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) : p : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) = r : ( ( real ) ( V11() real ext-real ) number ) } holds
P : ( ( ) ( ) Subset of ) is closed ;

theorem :: JORDAN6:7
for r being ( ( real ) ( V11() real ext-real ) number )
for P being ( ( ) ( ) Subset of ) st P : ( ( ) ( ) Subset of ) = { p : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) where p is ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) : p : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) >= r : ( ( real ) ( V11() real ext-real ) number ) } holds
P : ( ( ) ( ) Subset of ) is closed ;

theorem :: JORDAN6:8
for r being ( ( real ) ( V11() real ext-real ) number )
for P being ( ( ) ( ) Subset of ) st P : ( ( ) ( ) Subset of ) = { p : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) where p is ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) : p : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) <= r : ( ( real ) ( V11() real ext-real ) number ) } holds
P : ( ( ) ( ) Subset of ) is closed ;

theorem :: JORDAN6:9
for r being ( ( real ) ( V11() real ext-real ) number )
for P being ( ( ) ( ) Subset of ) st P : ( ( ) ( ) Subset of ) = { p : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) where p is ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) : p : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) = r : ( ( real ) ( V11() real ext-real ) number ) } holds
P : ( ( ) ( ) Subset of ) is closed ;

theorem :: JORDAN6:10
for n being ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) )
for P being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
P : ( ( ) ( ) Subset of ) is connected ;

theorem :: JORDAN6:11
for P being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
P : ( ( ) ( ) Subset of ) is closed ;

theorem :: JORDAN6:12
for P being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
ex q being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st
( q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) in P : ( ( ) ( ) Subset of ) & q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) = ((p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `1) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) + (p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `1) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) ;

theorem :: JORDAN6:13
for P, Q being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) & Q : ( ( ) ( ) Subset of ) = { q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) where q is ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) : q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) = ((p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `1) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) + (p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `1) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) } holds
( P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) & P : ( ( ) ( ) Subset of ) /\ Q : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) is closed ) ;

theorem :: JORDAN6:14
for P, Q being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) & Q : ( ( ) ( ) Subset of ) = { q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) where q is ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) : q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) = ((p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) + (p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) } holds
( P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) & P : ( ( ) ( ) Subset of ) /\ Q : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) is closed ) ;

definition
let P be ( ( ) ( ) Subset of ) ;
let p1, p2 be ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ;
func x_Middle (P,p1,p2) -> ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) means :: JORDAN6:def 1
 for Q being ( ( ) ( ) Subset of ) st Q : ( ( ) ( ) Subset of ) = { q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) where q is ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) : q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) = ((p1 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like total quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) `1) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) + (p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26(P : ( ( ) ( ) MetrStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) `1) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) } holds
it : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26(P : ( ( ) ( ) MetrStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = First_Point (P : ( ( ) ( ) MetrStruct ) ,p1 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like total quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26(P : ( ( ) ( ) MetrStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,Q : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;
end;

definition
let P be ( ( ) ( ) Subset of ) ;
let p1, p2 be ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ;
func y_Middle (P,p1,p2) -> ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) means :: JORDAN6:def 2
 for Q being ( ( ) ( ) Subset of ) st Q : ( ( ) ( ) Subset of ) = { q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) where q is ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) : q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) = ((p1 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like total quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) `2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) + (p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26(P : ( ( ) ( ) MetrStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) `2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) } holds
it : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26(P : ( ( ) ( ) MetrStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = First_Point (P : ( ( ) ( ) MetrStruct ) ,p1 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like total quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26(P : ( ( ) ( ) MetrStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,Q : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;
end;

theorem :: JORDAN6:15
for P being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
( x_Middle (P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) in P : ( ( ) ( ) Subset of ) & y_Middle (P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) in P : ( ( ) ( ) Subset of ) ) ;

theorem :: JORDAN6:16
for P being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
( p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) = x_Middle (P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) iff p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) = p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) ;

theorem :: JORDAN6:17
for P being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
( p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) = y_Middle (P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) iff p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) = p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) ;

begin

theorem :: JORDAN6:18
for P being ( ( ) ( ) Subset of )
for p1, p2, q1, q2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) & LE q1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
LE q2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( ) Subset of ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ;

definition
let P be ( ( ) ( ) Subset of ) ;
let p1, p2, q1 be ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ;
func L_Segment (P,p1,p2,q1) -> ( ( ) ( ) Subset of ) equals :: JORDAN6:def 3
 { q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) where q is ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) : LE q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q1 : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26(P : ( ( ) ( ) MetrStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,P : ( ( ) ( ) MetrStruct ) ,p1 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like total quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26(P : ( ( ) ( ) MetrStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) } ;
end;

definition
let P be ( ( ) ( ) Subset of ) ;
let p1, p2, q1 be ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ;
func R_Segment (P,p1,p2,q1) -> ( ( ) ( ) Subset of ) equals :: JORDAN6:def 4
 { q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) where q is ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) : LE q1 : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26(P : ( ( ) ( ) MetrStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( ) MetrStruct ) ,p1 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like total quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26(P : ( ( ) ( ) MetrStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) } ;
end;

theorem :: JORDAN6:19
for P being ( ( ) ( ) Subset of )
for p1, p2, q1 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds L_Segment (P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( ) Subset of ) c= P : ( ( ) ( ) Subset of ) ;

theorem :: JORDAN6:20
for P being ( ( ) ( ) Subset of )
for p1, p2, q1 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds R_Segment (P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( ) Subset of ) c= P : ( ( ) ( ) Subset of ) ;

theorem :: JORDAN6:21
for P being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
L_Segment (P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( ) Subset of ) = {p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty ) set ) ;

theorem :: JORDAN6:22
for P being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
L_Segment (P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( ) Subset of ) = P : ( ( ) ( ) Subset of ) ;

theorem :: JORDAN6:23
for P being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
R_Segment (P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( ) Subset of ) = {p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty ) set ) ;

theorem :: JORDAN6:24
for P being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
R_Segment (P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( ) Subset of ) = P : ( ( ) ( ) Subset of ) ;

theorem :: JORDAN6:25
for P being ( ( ) ( ) Subset of )
for p1, p2, q1 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
R_Segment (P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( ) Subset of ) = L_Segment (P : ( ( ) ( ) Subset of ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( ) Subset of ) ;

definition
let P be ( ( ) ( ) Subset of ) ;
let p1, p2, q1, q2 be ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ;
func Segment (P,p1,p2,q1,q2) -> ( ( ) ( ) Subset of ) equals :: JORDAN6:def 5
(R_Segment (P : ( ( ) ( ) MetrStruct ) ,p1 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like total quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26(P : ( ( ) ( ) MetrStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,q1 : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26(P : ( ( ) ( ) MetrStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( ) ( ) Subset of ) /\ (L_Segment (P : ( ( ) ( ) MetrStruct ) ,p1 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like total quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26(P : ( ( ) ( ) MetrStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,q2 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ;
end;

theorem :: JORDAN6:26
for P being ( ( ) ( ) Subset of )
for p1, p2, q1, q2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds Segment (P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( ) Subset of ) = { q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) where q is ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) : ( LE q1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) & LE q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ) } ;

theorem :: JORDAN6:27
for P being ( ( ) ( ) Subset of )
for p1, p2, q1, q2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) & LE q2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( ) Subset of ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
LE q1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN6:28
for P being ( ( ) ( ) Subset of )
for p1, p2, q being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
L_Segment (P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( ) Subset of ) = R_Segment (P : ( ( ) ( ) Subset of ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( ) Subset of ) ;

theorem :: JORDAN6:29
for P being ( ( ) ( ) Subset of )
for p1, p2, q1, q2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
Segment (P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( ) Subset of ) = Segment (P : ( ( ) ( ) Subset of ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( ) Subset of ) ;

begin

definition
let s be ( ( real ) ( V11() real ext-real ) number ) ;
func Vertical_Line s -> ( ( ) ( ) Subset of ) equals :: JORDAN6:def 6
 { p : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) where p is ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) : p : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) = s : ( ( ) ( ) MetrStruct ) } ;
func Horizontal_Line s -> ( ( ) ( ) Subset of ) equals :: JORDAN6:def 7
 { p : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) where p is ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) : p : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) = s : ( ( ) ( ) MetrStruct ) } ;
end;

theorem :: JORDAN6:30
for r being ( ( real ) ( V11() real ext-real ) number ) holds
( Vertical_Line r : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( ) Subset of ) is closed & Horizontal_Line r : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( ) Subset of ) is closed ) ;

theorem :: JORDAN6:31
for r being ( ( real ) ( V11() real ext-real ) number )
for p being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
( p : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) in Vertical_Line r : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( ) Subset of ) iff p : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) = r : ( ( real ) ( V11() real ext-real ) number ) ) ;

theorem :: JORDAN6:32
for r being ( ( real ) ( V11() real ext-real ) number )
for p being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
( p : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) in Horizontal_Line r : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( ) Subset of ) iff p : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) = r : ( ( real ) ( V11() real ext-real ) number ) ) ;

theorem :: JORDAN6:33
for P being ( ( ) ( ) Subset of ) st P : ( ( ) ( ) Subset of ) is being_simple_closed_curve holds
ex P1, P2 being ( ( non empty ) ( non empty ) Subset of ) st
( P1 : ( ( non empty ) ( non empty ) Subset of ) is_an_arc_of W-min P : ( ( ) ( ) Subset of ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) , E-max P : ( ( ) ( ) Subset of ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) & P2 : ( ( non empty ) ( non empty ) Subset of ) is_an_arc_of E-max P : ( ( ) ( ) Subset of ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) , W-min P : ( ( ) ( ) Subset of ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) & P1 : ( ( non empty ) ( non empty ) Subset of ) /\ P2 : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = {(W-min P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(E-max P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty ) set ) & P1 : ( ( non empty ) ( non empty ) Subset of ) \/ P2 : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( non empty ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = P : ( ( ) ( ) Subset of ) & (First_Point (P1 : ( ( non empty ) ( non empty ) Subset of ) ,(W-min P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(E-max P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(Vertical_Line (((W-bound P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) + (E-bound P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( ) Subset of ) )) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) > (Last_Point (P2 : ( ( non empty ) ( non empty ) Subset of ) ,(E-max P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(W-min P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(Vertical_Line (((W-bound P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) + (E-bound P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( ) Subset of ) )) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) ;

begin

theorem :: JORDAN6:34
for P being ( ( ) ( V146() V147() V148() ) Subset of ) st P : ( ( ) ( V146() V147() V148() ) Subset of ) = the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like T_0 T_1 T_2 compact V200() ) TopStruct ) : ( ( ) ( non empty V146() V147() V148() ) set ) \ {0 : ( ( ) ( empty natural V11() real ext-real non positive non negative V83() V84() V146() V147() V148() V149() V150() V151() V152() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) } : ( ( ) ( non empty V146() V147() V148() V149() V150() V151() ) set ) : ( ( ) ( V146() V147() V148() ) Element of K6( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like T_0 T_1 T_2 compact V200() ) TopStruct ) : ( ( ) ( non empty V146() V147() V148() ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( V146() V147() V148() ) Subset of ) is open ;

theorem :: JORDAN6:35
for P being ( ( ) ( V146() V147() V148() ) Subset of )
for r, s being ( ( real ) ( V11() real ext-real ) number ) st P : ( ( ) ( V146() V147() V148() ) Subset of ) = ].r : ( ( real ) ( V11() real ext-real ) number ) ,s : ( ( real ) ( V11() real ext-real ) number ) .[ : ( ( ) ( V146() V147() V148() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( V146() V147() V148() ) Subset of ) is open ;

theorem :: JORDAN6:36
for P7 being ( ( ) ( V146() V147() V148() ) Subset of ) st P7 : ( ( ) ( V146() V147() V148() ) Subset of ) = the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like T_0 T_1 T_2 compact V200() ) TopStruct ) : ( ( ) ( non empty V146() V147() V148() ) set ) \ {0 : ( ( ) ( empty natural V11() real ext-real non positive non negative V83() V84() V146() V147() V148() V149() V150() V151() V152() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) } : ( ( ) ( non empty V146() V147() V148() V149() V150() V151() ) set ) : ( ( ) ( V146() V147() V148() ) Element of K6( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like T_0 T_1 T_2 compact V200() ) TopStruct ) : ( ( ) ( non empty V146() V147() V148() ) set ) ) : ( ( ) ( ) set ) ) holds
( P7 : ( ( ) ( V146() V147() V148() ) Subset of ) <> {} : ( ( ) ( empty V146() V147() V148() V149() V150() V151() V152() ) set ) & P7 : ( ( ) ( V146() V147() V148() ) Subset of ) is connected ) ;

theorem :: JORDAN6:37
for n being ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) )
for P being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) <> p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN6:38
for n being ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) )
for P being ( ( ) ( ) Subset of )
for Q being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) & Q : ( ( ) ( ) Subset of ) = P : ( ( ) ( ) Subset of ) \ {p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty ) set ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) holds
Q : ( ( ) ( ) Subset of ) is open ;

theorem :: JORDAN6:39
for n being ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) )
for P, P1, P2 being ( ( ) ( ) Subset of )
for Q being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P2 : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) & P1 : ( ( ) ( ) Subset of ) \/ P2 : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = P : ( ( ) ( ) Subset of ) & P1 : ( ( ) ( ) Subset of ) /\ P2 : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = {p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty ) set ) & Q : ( ( ) ( ) Subset of ) = P1 : ( ( ) ( ) Subset of ) \ {p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty ) set ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) holds
Q : ( ( ) ( ) Subset of ) is open ;

theorem :: JORDAN6:40
for n being ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) )
for P being ( ( ) ( ) Subset of )
for Q being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) & Q : ( ( ) ( ) Subset of ) = P : ( ( ) ( ) Subset of ) \ {p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty ) set ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) holds
Q : ( ( ) ( ) Subset of ) is connected ;

theorem :: JORDAN6:41
for GX being ( ( TopSpace-like ) ( TopSpace-like ) TopSpace)
for P1, P being ( ( ) ( ) Subset of )
for Q9 being ( ( ) ( ) Subset of )
for Q being ( ( ) ( ) Subset of ) st P1 : ( ( ) ( ) Subset of ) c= P : ( ( ) ( ) Subset of ) & Q : ( ( ) ( ) Subset of ) = Q9 : ( ( ) ( ) Subset of ) & Q9 : ( ( ) ( ) Subset of ) is connected holds
Q : ( ( ) ( ) Subset of ) is connected ;

theorem :: JORDAN6:42
for n being ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) )
for P being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
ex p3 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st
( p3 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) in P : ( ( ) ( ) Subset of ) & p3 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) <> p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) & p3 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) <> p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ) ;

theorem :: JORDAN6:43
for n being ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) )
for P being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
P : ( ( ) ( ) Subset of ) \ {p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty ) set ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) <> {} : ( ( ) ( empty V146() V147() V148() V149() V150() V151() V152() ) set ) ;

theorem :: JORDAN6:44
for n being ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) )
for P1, P being ( ( ) ( ) Subset of )
for Q being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P1 : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) & P1 : ( ( ) ( ) Subset of ) c= P : ( ( ) ( ) Subset of ) & Q : ( ( ) ( ) Subset of ) = P1 : ( ( ) ( ) Subset of ) \ {p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty ) set ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) holds
Q : ( ( ) ( ) Subset of ) is connected ;

theorem :: JORDAN6:45
for T, S, V being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for P1 being ( ( non empty ) ( non empty ) Subset of )
for P2 being ( ( ) ( ) Subset of )
for f being ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V26( the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b4 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for g being ( ( Function-like quasi_total ) ( V22() V25( the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b5 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( ) set ) ) V26( the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty ) set ) ) st P1 : ( ( non empty ) ( non empty ) Subset of ) c= P2 : ( ( ) ( ) Subset of ) & f : ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V26( the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b4 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( V22() V25( the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b5 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( ) set ) ) V26( the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty ) set ) ) is continuous holds
ex h being ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V26( the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st
( h : ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V26( the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = g : ( ( Function-like quasi_total ) ( V22() V25( the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b5 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( ) set ) ) V26( the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V26( the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b4 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V22() V25( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V26( the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & h : ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V26( the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous ) ;

theorem :: JORDAN6:46
for n being ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) )
for P1, P2 being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P1 : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) & P2 : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(b1 : ( ( ) ( natural V11() real ext-real V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) & P1 : ( ( ) ( ) Subset of ) c= P2 : ( ( ) ( ) Subset of ) holds
P1 : ( ( ) ( ) Subset of ) = P2 : ( ( ) ( ) Subset of ) ;

theorem :: JORDAN6:47
for P being ( ( ) ( ) Subset of )
for Q being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is being_simple_closed_curve & p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) in P : ( ( ) ( ) Subset of ) & p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) in P : ( ( ) ( ) Subset of ) & p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) <> p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) & Q : ( ( ) ( ) Subset of ) = P : ( ( ) ( ) Subset of ) \ {p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty ) set ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) holds
not Q : ( ( ) ( ) Subset of ) is connected ;

theorem :: JORDAN6:48
for P, P1, P2, P19, P29 being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is being_simple_closed_curve & P1 : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) & P2 : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) & P1 : ( ( ) ( ) Subset of ) \/ P2 : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = P : ( ( ) ( ) Subset of ) & P19 : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) & P29 : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) & P19 : ( ( ) ( ) Subset of ) \/ P29 : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = P : ( ( ) ( ) Subset of ) & not ( P1 : ( ( ) ( ) Subset of ) = P19 : ( ( ) ( ) Subset of ) & P2 : ( ( ) ( ) Subset of ) = P29 : ( ( ) ( ) Subset of ) ) holds
( P1 : ( ( ) ( ) Subset of ) = P29 : ( ( ) ( ) Subset of ) & P2 : ( ( ) ( ) Subset of ) = P19 : ( ( ) ( ) Subset of ) ) ;

begin

theorem :: JORDAN6:49
for P1 being ( ( ) ( ) Subset of )
for r being ( ( real ) ( V11() real ext-real ) number )
for p1, p2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) <= r : ( ( real ) ( V11() real ext-real ) number ) & r : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) & P1 : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
( P1 : ( ( ) ( ) Subset of ) meets Vertical_Line r : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( ) Subset of ) & P1 : ( ( ) ( ) Subset of ) /\ (Vertical_Line r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) is closed ) ;

definition
let P be ( ( ) ( ) Subset of ) ;
assume P : ( ( ) ( ) Subset of ) is being_simple_closed_curve ;
func Upper_Arc P -> ( ( non empty ) ( non empty ) Subset of ) means :: JORDAN6:def 8
( it : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like total quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_an_arc_of W-min P : ( ( ) ( ) MetrStruct ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) , E-max P : ( ( ) ( ) MetrStruct ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) & ex P2 being ( ( non empty ) ( non empty ) Subset of ) st
( P2 : ( ( non empty ) ( non empty ) Subset of ) is_an_arc_of E-max P : ( ( ) ( ) MetrStruct ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) , W-min P : ( ( ) ( ) MetrStruct ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) & it : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like total quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /\ P2 : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = {(W-min P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(E-max P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty ) set ) & it : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like total quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) \/ P2 : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( non empty ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = P : ( ( ) ( ) MetrStruct ) & (First_Point (it : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like total quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,(W-min P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(E-max P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(Vertical_Line (((W-bound P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) + (E-bound P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( ) Subset of ) )) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) > (Last_Point (P2 : ( ( non empty ) ( non empty ) Subset of ) ,(E-max P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(W-min P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(Vertical_Line (((W-bound P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) + (E-bound P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( ) Subset of ) )) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) );
end;

definition
let P be ( ( ) ( ) Subset of ) ;
assume P : ( ( ) ( ) Subset of ) is being_simple_closed_curve ;
func Lower_Arc P -> ( ( non empty ) ( non empty ) Subset of ) means :: JORDAN6:def 9
( it : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like total quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_an_arc_of E-max P : ( ( ) ( ) MetrStruct ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) , W-min P : ( ( ) ( ) MetrStruct ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) & (Upper_Arc P : ( ( ) ( ) MetrStruct ) ) : ( ( non empty ) ( non empty ) Subset of ) /\ it : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like total quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = {(W-min P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(E-max P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty ) set ) & (Upper_Arc P : ( ( ) ( ) MetrStruct ) ) : ( ( non empty ) ( non empty ) Subset of ) \/ it : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like total quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = P : ( ( ) ( ) MetrStruct ) & (First_Point ((Upper_Arc P : ( ( ) ( ) MetrStruct ) ) : ( ( non empty ) ( non empty ) Subset of ) ,(W-min P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(E-max P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(Vertical_Line (((W-bound P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) + (E-bound P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( ) Subset of ) )) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) > (Last_Point (it : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like total quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,(E-max P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(W-min P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(Vertical_Line (((W-bound P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) + (E-bound P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( ) Subset of ) )) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) );
end;

theorem :: JORDAN6:50
for P being ( ( ) ( ) Subset of ) st P : ( ( ) ( ) Subset of ) is being_simple_closed_curve holds
( Upper_Arc P : ( ( ) ( ) Subset of ) : ( ( non empty ) ( non empty ) Subset of ) is_an_arc_of W-min P : ( ( ) ( ) Subset of ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) , E-max P : ( ( ) ( ) Subset of ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) & Upper_Arc P : ( ( ) ( ) Subset of ) : ( ( non empty ) ( non empty ) Subset of ) is_an_arc_of E-max P : ( ( ) ( ) Subset of ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) , W-min P : ( ( ) ( ) Subset of ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) & Lower_Arc P : ( ( ) ( ) Subset of ) : ( ( non empty ) ( non empty ) Subset of ) is_an_arc_of E-max P : ( ( ) ( ) Subset of ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) , W-min P : ( ( ) ( ) Subset of ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) & Lower_Arc P : ( ( ) ( ) Subset of ) : ( ( non empty ) ( non empty ) Subset of ) is_an_arc_of W-min P : ( ( ) ( ) Subset of ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) , E-max P : ( ( ) ( ) Subset of ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) & (Upper_Arc P : ( ( ) ( ) Subset of ) ) : ( ( non empty ) ( non empty ) Subset of ) /\ (Lower_Arc P : ( ( ) ( ) Subset of ) ) : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = {(W-min P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(E-max P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty ) set ) & (Upper_Arc P : ( ( ) ( ) Subset of ) ) : ( ( non empty ) ( non empty ) Subset of ) \/ (Lower_Arc P : ( ( ) ( ) Subset of ) ) : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( non empty ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = P : ( ( ) ( ) Subset of ) & (First_Point ((Upper_Arc P : ( ( ) ( ) Subset of ) ) : ( ( non empty ) ( non empty ) Subset of ) ,(W-min P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(E-max P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(Vertical_Line (((W-bound P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) + (E-bound P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( ) Subset of ) )) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) > (Last_Point ((Lower_Arc P : ( ( ) ( ) Subset of ) ) : ( ( non empty ) ( non empty ) Subset of ) ,(E-max P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(W-min P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(Vertical_Line (((W-bound P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) + (E-bound P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( ) Subset of ) )) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) ;

theorem :: JORDAN6:51
for P being ( ( ) ( ) Subset of ) st P : ( ( ) ( ) Subset of ) is being_simple_closed_curve holds
( Lower_Arc P : ( ( ) ( ) Subset of ) : ( ( non empty ) ( non empty ) Subset of ) = (P : ( ( ) ( ) Subset of ) \ (Upper_Arc P : ( ( ) ( ) Subset of ) ) : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) \/ {(W-min P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(E-max P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) & Upper_Arc P : ( ( ) ( ) Subset of ) : ( ( non empty ) ( non empty ) Subset of ) = (P : ( ( ) ( ) Subset of ) \ (Lower_Arc P : ( ( ) ( ) Subset of ) ) : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) \/ {(W-min P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(E-max P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: JORDAN6:52
for P being ( ( ) ( ) Subset of )
for P1 being ( ( ) ( ) Subset of ) st P : ( ( ) ( ) Subset of ) is being_simple_closed_curve & (Upper_Arc P : ( ( ) ( ) Subset of ) ) : ( ( non empty ) ( non empty ) Subset of ) /\ P1 : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) | b1 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = {(W-min P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(E-max P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty ) set ) & (Upper_Arc P : ( ( ) ( ) Subset of ) ) : ( ( non empty ) ( non empty ) Subset of ) \/ P1 : ( ( ) ( ) Subset of ) : ( ( ) ( non empty ) set ) = P : ( ( ) ( ) Subset of ) holds
P1 : ( ( ) ( ) Subset of ) = Lower_Arc P : ( ( ) ( ) Subset of ) : ( ( non empty ) ( non empty ) Subset of ) ;

theorem :: JORDAN6:53
for P being ( ( ) ( ) Subset of )
for P1 being ( ( ) ( ) Subset of ) st P : ( ( ) ( ) Subset of ) is being_simple_closed_curve & P1 : ( ( ) ( ) Subset of ) /\ (Lower_Arc P : ( ( ) ( ) Subset of ) ) : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) = {(W-min P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(E-max P : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty ) set ) & P1 : ( ( ) ( ) Subset of ) \/ (Lower_Arc P : ( ( ) ( ) Subset of ) ) : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( non empty ) set ) = P : ( ( ) ( ) Subset of ) holds
P1 : ( ( ) ( ) Subset of ) = Upper_Arc P : ( ( ) ( ) Subset of ) : ( ( non empty ) ( non empty ) Subset of ) ;

begin

theorem :: JORDAN6:54
for P being ( ( ) ( ) Subset of )
for p1, p2, q being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) & LE q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) = p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN6:55
for P being ( ( ) ( ) Subset of )
for p1, p2, q being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) & LE p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) = p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ;

definition
let P be ( ( ) ( ) Subset of ) ;
let q1, q2 be ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ;
pred LE q1,q2,P means :: JORDAN6:def 10
( ( q1 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like total quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) in Upper_Arc P : ( ( ) ( ) MetrStruct ) : ( ( non empty ) ( non empty ) Subset of ) & q2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26(P : ( ( ) ( ) MetrStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) in Lower_Arc P : ( ( ) ( ) MetrStruct ) : ( ( non empty ) ( non empty ) Subset of ) & not q2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26(P : ( ( ) ( ) MetrStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = W-min P : ( ( ) ( ) MetrStruct ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ) or ( q1 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like total quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) in Upper_Arc P : ( ( ) ( ) MetrStruct ) : ( ( non empty ) ( non empty ) Subset of ) & q2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26(P : ( ( ) ( ) MetrStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) in Upper_Arc P : ( ( ) ( ) MetrStruct ) : ( ( non empty ) ( non empty ) Subset of ) & LE q1 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like total quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,q2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26(P : ( ( ) ( ) MetrStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) , Upper_Arc P : ( ( ) ( ) MetrStruct ) : ( ( non empty ) ( non empty ) Subset of ) , W-min P : ( ( ) ( ) MetrStruct ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) , E-max P : ( ( ) ( ) MetrStruct ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ) or ( q1 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like total quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) in Lower_Arc P : ( ( ) ( ) MetrStruct ) : ( ( non empty ) ( non empty ) Subset of ) & q2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26(P : ( ( ) ( ) MetrStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) in Lower_Arc P : ( ( ) ( ) MetrStruct ) : ( ( non empty ) ( non empty ) Subset of ) & not q2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26(P : ( ( ) ( ) MetrStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = W-min P : ( ( ) ( ) MetrStruct ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) & LE q1 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like total quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,q2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) V26(P : ( ( ) ( ) MetrStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) MetrStruct ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) MetrStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) , Lower_Arc P : ( ( ) ( ) MetrStruct ) : ( ( non empty ) ( non empty ) Subset of ) , E-max P : ( ( ) ( ) MetrStruct ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) , W-min P : ( ( ) ( ) MetrStruct ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ) );
end;

theorem :: JORDAN6:56
for P being ( ( ) ( ) Subset of )
for q being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is being_simple_closed_curve & q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) in P : ( ( ) ( ) Subset of ) holds
LE q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( ) Subset of ) ;

theorem :: JORDAN6:57
for P being ( ( ) ( ) Subset of )
for q1, q2 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is being_simple_closed_curve & LE q1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( ) Subset of ) & LE q2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( ) Subset of ) holds
q1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) = q2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN6:58
for P being ( ( ) ( ) Subset of )
for q1, q2, q3 being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is being_simple_closed_curve & LE q1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( ) Subset of ) & LE q2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q3 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( ) Subset of ) holds
LE q1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q3 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( ) Subset of ) ;

theorem :: JORDAN6:59
for P being ( ( ) ( ) Subset of )
for p1, p2, q being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) & q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) <> p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
not p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) in L_Segment (P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( ) Subset of ) ;

theorem :: JORDAN6:60
for P being ( ( ) ( ) Subset of )
for p1, p2, q being ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) & q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) <> p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) holds
not p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) in R_Segment (P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ,q : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( ) Subset of ) ;

registration
let S be ( ( non empty being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Subset of ) ;
cluster Lower_Arc S : ( ( non empty being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) -> non empty compact ;
cluster Upper_Arc S : ( ( non empty being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty ) Subset of ) -> non empty compact ;
end;

theorem :: JORDAN6:61
for S being ( ( non empty being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Subset of ) holds
( Lower_Arc S : ( ( non empty being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Subset of ) : ( ( non empty ) ( non empty closed compact ) Subset of ) c= S : ( ( non empty being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Subset of ) & Upper_Arc S : ( ( non empty being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Subset of ) : ( ( non empty ) ( non empty closed compact ) Subset of ) c= S : ( ( non empty being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Subset of ) ) ;

definition
let C be ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) ;
func Lower_Middle_Point C -> ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) equals :: JORDAN6:def 11
 First_Point ((Lower_Arc C : ( ( non empty V220() ) ( non empty V220() ) L11()) ) : ( ( non empty ) ( non empty ) Subset of ) ,(W-min C : ( ( non empty V220() ) ( non empty V220() ) L11()) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(E-max C : ( ( non empty V220() ) ( non empty V220() ) L11()) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(Vertical_Line (((W-bound C : ( ( non empty V220() ) ( non empty V220() ) L11()) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) + (E-bound C : ( ( non empty V220() ) ( non empty V220() ) L11()) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;
func Upper_Middle_Point C -> ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) equals :: JORDAN6:def 12
 First_Point ((Upper_Arc C : ( ( non empty V220() ) ( non empty V220() ) L11()) ) : ( ( non empty ) ( non empty ) Subset of ) ,(W-min C : ( ( non empty V220() ) ( non empty V220() ) L11()) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(E-max C : ( ( non empty V220() ) ( non empty V220() ) L11()) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(Vertical_Line (((W-bound C : ( ( non empty V220() ) ( non empty V220() ) L11()) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) + (E-bound C : ( ( non empty V220() ) ( non empty V220() ) L11()) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( ) Subset of ) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;
end;

theorem :: JORDAN6:62
for C being ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) holds Lower_Arc C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) : ( ( non empty ) ( non empty closed compact ) Subset of ) meets Vertical_Line (((W-bound C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) + (E-bound C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) Subset of ) ;

theorem :: JORDAN6:63
for C being ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) holds Upper_Arc C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) : ( ( non empty ) ( non empty closed compact ) Subset of ) meets Vertical_Line (((W-bound C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) + (E-bound C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) Subset of ) ;

theorem :: JORDAN6:64
for C being ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) holds (Lower_Middle_Point C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) = ((W-bound C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) + (E-bound C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ;

theorem :: JORDAN6:65
for C being ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) holds (Upper_Middle_Point C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) ) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) = ((W-bound C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) + (E-bound C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ;

theorem :: JORDAN6:66
for C being ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) holds Lower_Middle_Point C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) in Lower_Arc C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) : ( ( non empty ) ( non empty closed compact ) Subset of ) ;

theorem :: JORDAN6:67
for C being ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) holds Upper_Middle_Point C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) in Upper_Arc C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) : ( ( non empty ) ( non empty closed compact ) Subset of ) ;

theorem :: JORDAN6:68
for C being ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) holds Upper_Middle_Point C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Point of ( ( ) ( non empty functional ) set ) ) in C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) ;

theorem :: JORDAN6:69
for C being ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve)
for r being ( ( real ) ( V11() real ext-real ) number ) st W-bound C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) <= r : ( ( real ) ( V11() real ext-real ) number ) & r : ( ( real ) ( V11() real ext-real ) number ) <= E-bound C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) holds
LSeg (|[r : ( ( real ) ( V11() real ext-real ) number ) ,(S-bound C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ]| : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,|[r : ( ( real ) ( V11() real ext-real ) number ) ,(N-bound C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ]| : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) meets Upper_Arc C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) : ( ( non empty ) ( non empty closed compact ) Subset of ) ;

theorem :: JORDAN6:70
for C being ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve)
for r being ( ( real ) ( V11() real ext-real ) number ) st W-bound C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) <= r : ( ( real ) ( V11() real ext-real ) number ) & r : ( ( real ) ( V11() real ext-real ) number ) <= E-bound C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) holds
LSeg (|[r : ( ( real ) ( V11() real ext-real ) number ) ,(S-bound C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ]| : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,|[r : ( ( real ) ( V11() real ext-real ) number ) ,(N-bound C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) ]| : ( ( ) ( V22() V26( REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) Function-like V43(2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V138() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V83() V84() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V146() V147() V148() V149() V150() V151() V152() V181() ) Element of K6(REAL : ( ( ) ( non empty V36() V146() V147() V148() V152() V181() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V112() V158() V159() V160() V161() V162() V163() V164() strict V231() V232() ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) meets Lower_Arc C : ( ( being_simple_closed_curve ) ( non empty closed compact being_simple_closed_curve ) Simple_closed_curve) : ( ( non empty ) ( non empty closed compact ) Subset of ) ;