:: JORDAN20 semantic presentation

begin

theorem :: JORDAN20:1
for P being ( ( ) ( ) Subset of )
for p1, p2, p being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) in P : ( ( ) ( ) Subset of ) holds
Segment (P : ( ( ) ( ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) ;

theorem :: JORDAN20:2
for p1, p2, p being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) )
for a being ( ( ) ( V11() real ext-real ) Real) st p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V226( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) <= a : ( ( ) ( V11() real ext-real ) Real) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) <= a : ( ( ) ( V11() real ext-real ) Real) holds
p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) <= a : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: JORDAN20:3
for p1, p2, p being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) )
for a being ( ( ) ( V11() real ext-real ) Real) st p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V226( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) >= a : ( ( ) ( V11() real ext-real ) Real) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) >= a : ( ( ) ( V11() real ext-real ) Real) holds
p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) >= a : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: JORDAN20:4
for p1, p2, p being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) )
for a being ( ( ) ( V11() real ext-real ) Real) st p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V226( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) < a : ( ( ) ( V11() real ext-real ) Real) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) < a : ( ( ) ( V11() real ext-real ) Real) holds
p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) < a : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: JORDAN20:5
for p1, p2, p being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) )
for a being ( ( ) ( V11() real ext-real ) Real) st p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V226( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) > a : ( ( ) ( V11() real ext-real ) Real) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) > a : ( ( ) ( V11() real ext-real ) Real) holds
p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) > a : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: JORDAN20:6
for j being ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2)
for p, q being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) < len f : ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2) : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) & p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) in LSeg (f : ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2) ,j : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & q : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) in LSeg (f : ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2) ,j : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & (f : ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2) /. j : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) = (f : ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2) /. (j : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) & (f : ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2) /. j : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) > (f : ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2) /. (j : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) & LE p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) , L~ f : ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,f : ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2) /. 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,f : ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2) /. (len f : ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) holds
p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) >= q : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) ;

theorem :: JORDAN20:7
for j being ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2)
for p, q being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) < len f : ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2) : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) & p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) in LSeg (f : ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2) ,j : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & q : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) in LSeg (f : ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2) ,j : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & (f : ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2) /. j : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) = (f : ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2) /. (j : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) & (f : ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2) /. j : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) < (f : ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2) /. (j : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) & LE p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) , L~ f : ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,f : ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2) /. 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,f : ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2) /. (len f : ( ( being_S-Seq ) ( V22() V25( NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) V26( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like FinSequence-like being_S-Seq ) S-Sequence_in_R2) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) holds
p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) <= q : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) ;

definition
let P be ( ( ) ( ) Subset of ) ;
let p1, p2, p be ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ;
let e be ( ( ) ( V11() real ext-real ) Real) ;
pred p is_Lin P,p1,p2,e means :: JORDAN20:def 1
( P : ( ( ) ( ) TopStruct ) is_an_arc_of p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) in P : ( ( ) ( ) TopStruct ) & p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) = e : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ex p4 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st
( p4 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) < e : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & LE p4 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & ( for p5 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st LE p4 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p5 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & LE p5 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) holds
p5 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) <= e : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) );
pred p is_Rin P,p1,p2,e means :: JORDAN20:def 2
( P : ( ( ) ( ) TopStruct ) is_an_arc_of p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) in P : ( ( ) ( ) TopStruct ) & p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) = e : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ex p4 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st
( p4 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) > e : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & LE p4 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & ( for p5 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st LE p4 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p5 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & LE p5 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) holds
p5 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) >= e : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) );
pred p is_Lout P,p1,p2,e means :: JORDAN20:def 3
( P : ( ( ) ( ) TopStruct ) is_an_arc_of p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) in P : ( ( ) ( ) TopStruct ) & p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) = e : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ex p4 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st
( p4 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) < e : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & LE p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & ( for p5 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st LE p5 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & LE p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) ,p5 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) holds
p5 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) <= e : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) );
pred p is_Rout P,p1,p2,e means :: JORDAN20:def 4
( P : ( ( ) ( ) TopStruct ) is_an_arc_of p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) in P : ( ( ) ( ) TopStruct ) & p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) = e : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ex p4 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st
( p4 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) > e : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & LE p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & ( for p5 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st LE p5 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & LE p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) ,p5 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) holds
p5 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) >= e : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) );
pred p is_OSin P,p1,p2,e means :: JORDAN20:def 5
( P : ( ( ) ( ) TopStruct ) is_an_arc_of p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) in P : ( ( ) ( ) TopStruct ) & p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) = e : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ex p7 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st
( LE p7 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & ( for p8 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st LE p7 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p8 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & LE p8 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) holds
p8 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) = e : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) & ( for p4 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st LE p4 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p7 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) <> p7 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) holds
( ex p5 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st
( LE p4 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p5 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & LE p5 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p7 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & p5 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) > e : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) & ex p6 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st
( LE p4 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p6 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & LE p6 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p7 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & p6 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) < e : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ) );
pred p is_OSout P,p1,p2,e means :: JORDAN20:def 6
( P : ( ( ) ( ) TopStruct ) is_an_arc_of p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) in P : ( ( ) ( ) TopStruct ) & p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) = e : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ex p7 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st
( LE p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) ,p7 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & ( for p8 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st LE p8 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p7 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & LE p : ( ( Function-like quasi_total ) ( V22() V25(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) ,p8 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) holds
p8 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) = e : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) & ( for p4 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st LE p7 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) <> p7 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) holds
( ex p5 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st
( LE p5 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & LE p7 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p5 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & p5 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) > e : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) & ex p6 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st
( LE p6 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & LE p7 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p6 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( ) ( ) TopStruct ) ,p1 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,p2 : ( ( Function-like quasi_total ) ( V22() V25(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) V26(P : ( ( ) ( ) TopStruct ) ) Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) TopStruct ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ,P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V22() ) set ) ) : ( ( ) ( ) set ) ) & p6 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) < e : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ) );
end;

theorem :: JORDAN20:8
for P being ( ( ) ( ) Subset of )
for p1, p2, p being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) )
for e being ( ( ) ( V11() real ext-real ) Real) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) <= e : ( ( ) ( V11() real ext-real ) Real) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) >= e : ( ( ) ( V11() real ext-real ) Real) holds
ex p3 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st
( p3 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) in P : ( ( ) ( ) Subset of ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) = e : ( ( ) ( V11() real ext-real ) Real) ) ;

theorem :: JORDAN20:9
for P being ( ( non empty ) ( non empty ) Subset of )
for p1, p2, p being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) )
for e being ( ( ) ( V11() real ext-real ) Real) st P : ( ( non empty ) ( non empty ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) < e : ( ( ) ( V11() real ext-real ) Real) & p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) in P : ( ( non empty ) ( non empty ) Subset of ) & p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) = e : ( ( ) ( V11() real ext-real ) Real) & not p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) is_Lin P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,e : ( ( ) ( V11() real ext-real ) Real) & not p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) is_Rin P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,e : ( ( ) ( V11() real ext-real ) Real) holds
p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) is_OSin P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,e : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: JORDAN20:10
for P being ( ( non empty ) ( non empty ) Subset of )
for p1, p2, p being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) )
for e being ( ( ) ( V11() real ext-real ) Real) st P : ( ( non empty ) ( non empty ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) > e : ( ( ) ( V11() real ext-real ) Real) & p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) in P : ( ( non empty ) ( non empty ) Subset of ) & p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) = e : ( ( ) ( V11() real ext-real ) Real) & not p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) is_Lout P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,e : ( ( ) ( V11() real ext-real ) Real) & not p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) is_Rout P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,e : ( ( ) ( V11() real ext-real ) Real) holds
p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) is_OSout P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,e : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: JORDAN20:11
for P being ( ( ) ( V143() V144() V145() ) Subset of )
for s being ( ( ) ( V11() real ext-real ) Real) st P : ( ( ) ( V143() V144() V145() ) Subset of ) = [.0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V22() non-empty empty-yielding V80() V81() V143() V144() V145() V146() V147() V148() V149() bounded_below V209() ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ,s : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V143() V144() V145() non right_end V209() ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( V143() V144() V145() ) Subset of ) is open ;

theorem :: JORDAN20:12
for P being ( ( ) ( V143() V144() V145() ) Subset of )
for s being ( ( ) ( V11() real ext-real ) Real) st P : ( ( ) ( V143() V144() V145() ) Subset of ) = ].s : ( ( ) ( V11() real ext-real ) Real) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( V143() V144() V145() non left_end V209() ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( V143() V144() V145() ) Subset of ) is open ;

theorem :: JORDAN20:13
for P being ( ( non empty ) ( non empty ) Subset of )
for P1 being ( ( ) ( ) Subset of )
for Q being ( ( ) ( V143() V144() V145() ) Subset of )
for f being ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like V200() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V200() ) TopStruct ) ) : ( ( ) ( non empty V143() V144() V145() ) set ) ) V26( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) | b1 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V143() V144() V145() ) set ) , ( ( ) ( non empty ) set ) )
for s being ( ( ) ( V11() real ext-real ) Real) st s : ( ( ) ( V11() real ext-real ) Real) <= 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) & P1 : ( ( ) ( ) Subset of ) = { q0 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) where q0 is ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) : ex ss being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V22() non-empty empty-yielding V80() V81() V143() V144() V145() V146() V147() V148() V149() bounded_below V209() ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) <= ss : ( ( ) ( V11() real ext-real ) Real) & ss : ( ( ) ( V11() real ext-real ) Real) < s : ( ( ) ( V11() real ext-real ) Real) & q0 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) = f : ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like V200() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V200() ) TopStruct ) ) : ( ( ) ( non empty V143() V144() V145() ) set ) ) V26( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) | b1 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V143() V144() V145() ) set ) , ( ( ) ( non empty ) set ) ) . ss : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( ) set ) ) } & Q : ( ( ) ( V143() V144() V145() ) Subset of ) = [.0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V22() non-empty empty-yielding V80() V81() V143() V144() V145() V146() V147() V148() V149() bounded_below V209() ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ,s : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V143() V144() V145() non right_end V209() ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like V200() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V200() ) TopStruct ) ) : ( ( ) ( non empty V143() V144() V145() ) set ) ) V26( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) | b1 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V143() V144() V145() ) set ) , ( ( ) ( non empty ) set ) ) .: Q : ( ( ) ( V143() V144() V145() ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) | b1 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = P1 : ( ( ) ( ) Subset of ) ;

theorem :: JORDAN20:14
for P being ( ( non empty ) ( non empty ) Subset of )
for P1 being ( ( ) ( ) Subset of )
for Q being ( ( ) ( V143() V144() V145() ) Subset of )
for f being ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like V200() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V200() ) TopStruct ) ) : ( ( ) ( non empty V143() V144() V145() ) set ) ) V26( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) | b1 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V143() V144() V145() ) set ) , ( ( ) ( non empty ) set ) )
for s being ( ( ) ( V11() real ext-real ) Real) st s : ( ( ) ( V11() real ext-real ) Real) >= 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V22() non-empty empty-yielding V80() V81() V143() V144() V145() V146() V147() V148() V149() bounded_below V209() ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) & P1 : ( ( ) ( ) Subset of ) = { q0 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) where q0 is ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) : ex ss being ( ( ) ( V11() real ext-real ) Real) st
( s : ( ( ) ( V11() real ext-real ) Real) < ss : ( ( ) ( V11() real ext-real ) Real) & ss : ( ( ) ( V11() real ext-real ) Real) <= 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) & q0 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) = f : ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like V200() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V200() ) TopStruct ) ) : ( ( ) ( non empty V143() V144() V145() ) set ) ) V26( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) | b1 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V143() V144() V145() ) set ) , ( ( ) ( non empty ) set ) ) . ss : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( ) set ) ) } & Q : ( ( ) ( V143() V144() V145() ) Subset of ) = ].s : ( ( ) ( V11() real ext-real ) Real) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( V143() V144() V145() non left_end V209() ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like V200() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V200() ) TopStruct ) ) : ( ( ) ( non empty V143() V144() V145() ) set ) ) V26( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) | b1 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V143() V144() V145() ) set ) , ( ( ) ( non empty ) set ) ) .: Q : ( ( ) ( V143() V144() V145() ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) | b1 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = P1 : ( ( ) ( ) Subset of ) ;

theorem :: JORDAN20:15
for P being ( ( non empty ) ( non empty ) Subset of )
for P1 being ( ( ) ( ) Subset of )
for f being ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like V200() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V200() ) TopStruct ) ) : ( ( ) ( non empty V143() V144() V145() ) set ) ) V26( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) | b1 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V143() V144() V145() ) set ) , ( ( ) ( non empty ) set ) )
for s being ( ( ) ( V11() real ext-real ) Real) st s : ( ( ) ( V11() real ext-real ) Real) <= 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like V200() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V200() ) TopStruct ) ) : ( ( ) ( non empty V143() V144() V145() ) set ) ) V26( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) | b1 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V143() V144() V145() ) set ) , ( ( ) ( non empty ) set ) ) is being_homeomorphism & P1 : ( ( ) ( ) Subset of ) = { q0 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) where q0 is ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) : ex ss being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V22() non-empty empty-yielding V80() V81() V143() V144() V145() V146() V147() V148() V149() bounded_below V209() ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) <= ss : ( ( ) ( V11() real ext-real ) Real) & ss : ( ( ) ( V11() real ext-real ) Real) < s : ( ( ) ( V11() real ext-real ) Real) & q0 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) = f : ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like V200() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V200() ) TopStruct ) ) : ( ( ) ( non empty V143() V144() V145() ) set ) ) V26( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) | b1 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V143() V144() V145() ) set ) , ( ( ) ( non empty ) set ) ) . ss : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( ) set ) ) } holds
P1 : ( ( ) ( ) Subset of ) is open ;

theorem :: JORDAN20:16
for P being ( ( non empty ) ( non empty ) Subset of )
for P1 being ( ( ) ( ) Subset of )
for f being ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like V200() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V200() ) TopStruct ) ) : ( ( ) ( non empty V143() V144() V145() ) set ) ) V26( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) | b1 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V143() V144() V145() ) set ) , ( ( ) ( non empty ) set ) )
for s being ( ( ) ( V11() real ext-real ) Real) st s : ( ( ) ( V11() real ext-real ) Real) >= 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V22() non-empty empty-yielding V80() V81() V143() V144() V145() V146() V147() V148() V149() bounded_below V209() ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like V200() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V200() ) TopStruct ) ) : ( ( ) ( non empty V143() V144() V145() ) set ) ) V26( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) | b1 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V143() V144() V145() ) set ) , ( ( ) ( non empty ) set ) ) is being_homeomorphism & P1 : ( ( ) ( ) Subset of ) = { q0 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) where q0 is ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) : ex ss being ( ( ) ( V11() real ext-real ) Real) st
( s : ( ( ) ( V11() real ext-real ) Real) < ss : ( ( ) ( V11() real ext-real ) Real) & ss : ( ( ) ( V11() real ext-real ) Real) <= 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) & q0 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) = f : ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like V200() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V200() ) TopStruct ) ) : ( ( ) ( non empty V143() V144() V145() ) set ) ) V26( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) | b1 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V143() V144() V145() ) set ) , ( ( ) ( non empty ) set ) ) . ss : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( ) set ) ) } holds
P1 : ( ( ) ( ) Subset of ) is open ;

theorem :: JORDAN20:17
for T being ( ( non empty ) ( non empty ) TopStruct )
for Q1, Q2 being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st Q1 : ( ( ) ( ) Subset of ) /\ Q2 : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty ) ( non empty ) TopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {} : ( ( ) ( empty V22() non-empty empty-yielding V143() V144() V145() V146() V147() V148() V149() bounded_below V209() ) set ) & Q1 : ( ( ) ( ) Subset of ) \/ Q2 : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty ) ( non empty ) TopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = the carrier of T : ( ( non empty ) ( non empty ) TopStruct ) : ( ( ) ( non empty ) set ) & p1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in Q1 : ( ( ) ( ) Subset of ) & p2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in Q2 : ( ( ) ( ) Subset of ) & Q1 : ( ( ) ( ) Subset of ) is open & Q2 : ( ( ) ( ) Subset of ) is open holds
for P being ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like V200() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V200() ) TopStruct ) ) : ( ( ) ( non empty V143() V144() V145() ) set ) ) V26( the carrier of b1 : ( ( non empty ) ( non empty ) TopStruct ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V143() V144() V145() ) set ) , ( ( ) ( non empty ) set ) ) holds
( not P : ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like V200() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V200() ) TopStruct ) ) : ( ( ) ( non empty V143() V144() V145() ) set ) ) V26( the carrier of b1 : ( ( non empty ) ( non empty ) TopStruct ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V143() V144() V145() ) set ) , ( ( ) ( non empty ) set ) ) is continuous or not P : ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like V200() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V200() ) TopStruct ) ) : ( ( ) ( non empty V143() V144() V145() ) set ) ) V26( the carrier of b1 : ( ( non empty ) ( non empty ) TopStruct ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V143() V144() V145() ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V22() non-empty empty-yielding V80() V81() V143() V144() V145() V146() V147() V148() V149() bounded_below V209() ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) or not P : ( ( Function-like quasi_total ) ( non empty V22() V25( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like V200() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V200() ) TopStruct ) ) : ( ( ) ( non empty V143() V144() V145() ) set ) ) V26( the carrier of b1 : ( ( non empty ) ( non empty ) TopStruct ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V143() V144() V145() ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: JORDAN20:18
for P being ( ( non empty ) ( non empty ) Subset of )
for Q being ( ( ) ( ) Subset of )
for p1, p2, q being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st P : ( ( non empty ) ( non empty ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) & q : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) in P : ( ( non empty ) ( non empty ) Subset of ) & q : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) <> p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) & q : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) <> p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) = P : ( ( non empty ) ( non empty ) Subset of ) \ {q : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
( not Q : ( ( ) ( ) Subset of ) is connected & ( for R being ( ( Function-like quasi_total ) ( V22() V25( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like V200() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V200() ) TopStruct ) ) : ( ( ) ( non empty V143() V144() V145() ) set ) ) V26( the carrier of (((TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) | b1 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) | b2 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) | b1 : ( ( non empty ) ( non empty ) Subset of ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) ) : ( ( ) ( ) set ) ) Function-like quasi_total ) Function of ( ( ) ( non empty V143() V144() V145() ) set ) , ( ( ) ( ) set ) ) holds
( not R : ( ( Function-like quasi_total ) ( V22() V25( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like V200() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V200() ) TopStruct ) ) : ( ( ) ( non empty V143() V144() V145() ) set ) ) V26( the carrier of (((TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) | b1 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) | b2 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) | b1 : ( ( non empty ) ( non empty ) Subset of ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) ) : ( ( ) ( ) set ) ) Function-like quasi_total ) Function of ( ( ) ( non empty V143() V144() V145() ) set ) , ( ( ) ( ) set ) ) is continuous or not R : ( ( Function-like quasi_total ) ( V22() V25( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like V200() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V200() ) TopStruct ) ) : ( ( ) ( non empty V143() V144() V145() ) set ) ) V26( the carrier of (((TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) | b1 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) | b2 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) | b1 : ( ( non empty ) ( non empty ) Subset of ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) ) : ( ( ) ( ) set ) ) Function-like quasi_total ) Function of ( ( ) ( non empty V143() V144() V145() ) set ) , ( ( ) ( ) set ) ) . 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V22() non-empty empty-yielding V80() V81() V143() V144() V145() V146() V147() V148() V149() bounded_below V209() ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) or not R : ( ( Function-like quasi_total ) ( V22() V25( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like V200() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V200() ) TopStruct ) ) : ( ( ) ( non empty V143() V144() V145() ) set ) ) V26( the carrier of (((TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) | b1 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) | b2 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) | b1 : ( ( non empty ) ( non empty ) Subset of ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) ) : ( ( ) ( ) set ) ) Function-like quasi_total ) Function of ( ( ) ( non empty V143() V144() V145() ) set ) , ( ( ) ( ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ) ) ) ;

theorem :: JORDAN20:19
for P being ( ( non empty ) ( non empty ) Subset of )
for p1, p2, q1, q2 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st P : ( ( non empty ) ( non empty ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) & q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) in P : ( ( non empty ) ( non empty ) Subset of ) & q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) in P : ( ( non empty ) ( non empty ) Subset of ) & not LE q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) holds
LE q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: JORDAN20:20
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) )
for p1, p2 being ( ( ) ( V43(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) )
for P, P1 being ( ( non empty ) ( non empty ) Subset of ) st P : ( ( non empty ) ( non empty ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) & P1 : ( ( non empty ) ( non empty ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V80() V81() V143() V144() V145() V146() V147() V148() bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) & P1 : ( ( non empty ) ( non empty ) Subset of ) c= P : ( ( non empty ) ( non empty ) Subset of ) holds
P1 : ( ( non empty ) ( non empty ) Subset of ) = P : ( ( non empty ) ( non empty ) Subset of ) ;

theorem :: JORDAN20:21
for P being ( ( non empty ) ( non empty ) Subset of )
for p1, p2, q1 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st P : ( ( non empty ) ( non empty ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) & q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) in P : ( ( non empty ) ( non empty ) Subset of ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) <> q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) holds
Segment (P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is_an_arc_of q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: JORDAN20:22
for P being ( ( non empty ) ( non empty ) Subset of )
for p1, p2, q1, q2, q3 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st P : ( ( non empty ) ( non empty ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) & LE q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) & LE q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q3 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) holds
(Segment (P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) \/ (Segment (P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q3 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = Segment (P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q3 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN20:23
for P being ( ( non empty ) ( non empty ) Subset of )
for p1, p2, q1, q2, q3 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st P : ( ( non empty ) ( non empty ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) & LE q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) & LE q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q3 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) holds
(Segment (P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (Segment (P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q3 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) ;

theorem :: JORDAN20:24
for P being ( ( non empty ) ( non empty ) Subset of )
for p1, p2 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st P : ( ( non empty ) ( non empty ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) holds
Segment (P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = P : ( ( non empty ) ( non empty ) Subset of ) ;

theorem :: JORDAN20:25
for P, Q1 being ( ( non empty ) ( non empty ) Subset of )
for p1, p2, q1, q2 being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) st P : ( ( non empty ) ( non empty ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) & Q1 : ( ( non empty ) ( non empty ) Subset of ) is_an_arc_of q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) & LE q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) & Q1 : ( ( non empty ) ( non empty ) Subset of ) c= P : ( ( non empty ) ( non empty ) Subset of ) holds
Q1 : ( ( non empty ) ( non empty ) Subset of ) = Segment (P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN20:26
for P being ( ( non empty ) ( non empty ) Subset of )
for p1, p2, q1, q2, p being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) )
for e being ( ( ) ( V11() real ext-real ) Real) st q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) is_Lin P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,e : ( ( ) ( V11() real ext-real ) Real) & q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) = e : ( ( ) ( V11() real ext-real ) Real) & LSeg (q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V226( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= P : ( ( non empty ) ( non empty ) Subset of ) & p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) in LSeg (q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V226( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) is_Lin P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,e : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: JORDAN20:27
for P being ( ( non empty ) ( non empty ) Subset of )
for p1, p2, q1, q2, p being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) )
for e being ( ( ) ( V11() real ext-real ) Real) st q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) is_Rin P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,e : ( ( ) ( V11() real ext-real ) Real) & q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) = e : ( ( ) ( V11() real ext-real ) Real) & LSeg (q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V226( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= P : ( ( non empty ) ( non empty ) Subset of ) & p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) in LSeg (q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V226( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) is_Rin P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,e : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: JORDAN20:28
for P being ( ( non empty ) ( non empty ) Subset of )
for p1, p2, q1, q2, p being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) )
for e being ( ( ) ( V11() real ext-real ) Real) st q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) is_Lout P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,e : ( ( ) ( V11() real ext-real ) Real) & q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) = e : ( ( ) ( V11() real ext-real ) Real) & LSeg (q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V226( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= P : ( ( non empty ) ( non empty ) Subset of ) & p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) in LSeg (q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V226( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) is_Lout P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,e : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: JORDAN20:29
for P being ( ( non empty ) ( non empty ) Subset of )
for p1, p2, q1, q2, p being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) )
for e being ( ( ) ( V11() real ext-real ) Real) st q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) is_Rout P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,e : ( ( ) ( V11() real ext-real ) Real) & q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) = e : ( ( ) ( V11() real ext-real ) Real) & LSeg (q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V226( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= P : ( ( non empty ) ( non empty ) Subset of ) & p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) in LSeg (q1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V226( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V109() V155() V156() V157() V158() V159() V160() V161() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) is_Rout P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,e : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: JORDAN20:30
for P being ( ( non empty ) ( non empty ) Subset of )
for p1, p2, p being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) )
for e being ( ( ) ( V11() real ext-real ) Real) st P : ( ( non empty ) ( non empty ) Subset of ) is_S-P_arc_joining p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) < e : ( ( ) ( V11() real ext-real ) Real) & p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) in P : ( ( non empty ) ( non empty ) Subset of ) & p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) = e : ( ( ) ( V11() real ext-real ) Real) & not p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) is_Lin P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,e : ( ( ) ( V11() real ext-real ) Real) holds
p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) is_Rin P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,e : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: JORDAN20:31
for P being ( ( non empty ) ( non empty ) Subset of )
for p1, p2, p being ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) )
for e being ( ( ) ( V11() real ext-real ) Real) st P : ( ( non empty ) ( non empty ) Subset of ) is_S-P_arc_joining p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) > e : ( ( ) ( V11() real ext-real ) Real) & p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) in P : ( ( non empty ) ( non empty ) Subset of ) & p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) = e : ( ( ) ( V11() real ext-real ) Real) & not p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) is_Lout P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,e : ( ( ) ( V11() real ext-real ) Real) holds
p : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) is_Rout P : ( ( non empty ) ( non empty ) Subset of ) ,p1 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V80() V81() V143() V144() V145() V146() V147() V148() left_end bounded_below ) Element of NAT : ( ( ) ( V143() V144() V145() V146() V147() V148() V149() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V36() V143() V144() V145() V149() non bounded_below non bounded_above V209() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V135() ) Point of ( ( ) ( non empty ) set ) ) ,e : ( ( ) ( V11() real ext-real ) Real) ;