:: TOPREAL3 semantic presentation

begin

begin

theorem :: TOPREAL3:1
for x, y, z being ( ( ) ( ) set ) holds
( 1 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) in dom <*x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ,z : ( ( ) ( ) set ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(3 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) : ( ( ) ( V137() V138() V139() V140() V141() V142() ) Element of K19(NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) & 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) in dom <*x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ,z : ( ( ) ( ) set ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(3 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) : ( ( ) ( V137() V138() V139() V140() V141() V142() ) Element of K19(NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) & 3 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) in dom <*x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ,z : ( ( ) ( ) set ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(3 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) : ( ( ) ( V137() V138() V139() V140() V141() V142() ) Element of K19(NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: TOPREAL3:2
for p1, p2 being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) holds
( (p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) + p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = (p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) + (p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & (p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) + p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = (p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) + (p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) ;

theorem :: TOPREAL3:3
for p1, p2 being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) holds
( (p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = (p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) - (p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & (p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = (p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) - (p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) ;

theorem :: TOPREAL3:4
for p being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) )
for r being ( ( real ) ( V22() real ext-real ) number ) holds
( (r : ( ( real ) ( V22() real ext-real ) number ) * p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = r : ( ( real ) ( V22() real ext-real ) number ) * (p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & (r : ( ( real ) ( V22() real ext-real ) number ) * p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = r : ( ( real ) ( V22() real ext-real ) number ) * (p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) ;

theorem :: TOPREAL3:5
for p1, p2 being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) )
for r1, s1, r2, s2 being ( ( real ) ( V22() real ext-real ) number ) st p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) = <*r1 : ( ( real ) ( V22() real ext-real ) number ) ,s1 : ( ( real ) ( V22() real ext-real ) number ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) & p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) = <*r2 : ( ( real ) ( V22() real ext-real ) number ) ,s2 : ( ( real ) ( V22() real ext-real ) number ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) holds
( p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) + p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) = <*(r1 : ( ( real ) ( V22() real ext-real ) number ) + r2 : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) set ) ,(s1 : ( ( real ) ( V22() real ext-real ) number ) + s2 : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) set ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) & p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) - p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) = <*(r1 : ( ( real ) ( V22() real ext-real ) number ) - r2 : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) set ) ,(s1 : ( ( real ) ( V22() real ext-real ) number ) - s2 : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) set ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) ) ;

theorem :: TOPREAL3:6
for p, q being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) holds
p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) = q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TOPREAL3:7
for p1, p2 being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) )
for u1, u2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st u1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) & u2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) holds
(Pitag_dist 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V30(K20((REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) ( Relation-like K20((REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) -valued Function-like V30(K20((REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) complex-yielding V128() V129() ) Element of K19(K20(K20((REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( complex-yielding V128() V129() ) set ) ) : ( ( ) ( ) set ) ) . (u1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,u2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) = sqrt ((((p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) - (p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ^2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) + (((p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) - (p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ^2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ;

theorem :: TOPREAL3:8
for n being ( ( natural ) ( ordinal natural V22() real ext-real non negative ) Nat) holds the carrier of (TOP-REAL n : ( ( natural ) ( ordinal natural V22() real ext-real non negative ) Nat) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) = the carrier of (Euclid n : ( ( natural ) ( ordinal natural V22() real ext-real non negative ) Nat) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ;

theorem :: TOPREAL3:9
for r1, s1, r being ( ( real ) ( V22() real ext-real ) number ) st r1 : ( ( real ) ( V22() real ext-real ) number ) <= s1 : ( ( real ) ( V22() real ext-real ) number ) holds
{ p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) where p1 is ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) : ( p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = r : ( ( real ) ( V22() real ext-real ) number ) & r1 : ( ( real ) ( V22() real ext-real ) number ) <= p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) <= s1 : ( ( real ) ( V22() real ext-real ) number ) ) } = LSeg (|[r : ( ( real ) ( V22() real ext-real ) number ) ,r1 : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,|[r : ( ( real ) ( V22() real ext-real ) number ) ,s1 : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL3:10
for r1, s1, r being ( ( real ) ( V22() real ext-real ) number ) st r1 : ( ( real ) ( V22() real ext-real ) number ) <= s1 : ( ( real ) ( V22() real ext-real ) number ) holds
{ p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) where p1 is ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) : ( p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = r : ( ( real ) ( V22() real ext-real ) number ) & r1 : ( ( real ) ( V22() real ext-real ) number ) <= p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) <= s1 : ( ( real ) ( V22() real ext-real ) number ) ) } = LSeg (|[r1 : ( ( real ) ( V22() real ext-real ) number ) ,r : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,|[s1 : ( ( real ) ( V22() real ext-real ) number ) ,r : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL3:11
for p being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) )
for r, r1, s1 being ( ( real ) ( V22() real ext-real ) number ) st p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) in LSeg (|[r : ( ( real ) ( V22() real ext-real ) number ) ,r1 : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,|[r : ( ( real ) ( V22() real ext-real ) number ) ,s1 : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = r : ( ( real ) ( V22() real ext-real ) number ) ;

theorem :: TOPREAL3:12
for p being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) )
for r1, r, s1 being ( ( real ) ( V22() real ext-real ) number ) st p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) in LSeg (|[r1 : ( ( real ) ( V22() real ext-real ) number ) ,r : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,|[s1 : ( ( real ) ( V22() real ext-real ) number ) ,r : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = r : ( ( real ) ( V22() real ext-real ) number ) ;

theorem :: TOPREAL3:13
for p, q being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) <> q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) holds
|[(((p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) + (q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in LSeg (p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL3:14
for p, q being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) <> q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) holds
|[(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(((p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) + (q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in LSeg (p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL3:15
for p, p1, q being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) )
for i, j being ( ( natural ) ( ordinal natural V22() real ext-real non negative ) Nat) st f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) = <*p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(3 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) & i : ( ( natural ) ( ordinal natural V22() real ext-real non negative ) Nat) <> 0 : ( ( ) ( functional empty ordinal natural V22() real ext-real non positive non negative V33() V34() FinSequence-membered V137() V138() V139() V140() V141() V142() V143() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( natural ) ( ordinal natural V22() real ext-real non negative ) Nat) > i : ( ( natural ) ( ordinal natural V22() real ext-real non negative ) Nat) + 1 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) holds
LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ,j : ( ( natural ) ( ordinal natural V22() real ext-real non negative ) Nat) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {} : ( ( ) ( functional empty FinSequence-membered V137() V138() V139() V140() V141() V142() V143() ) set ) ;

theorem :: TOPREAL3:16
for p1, p2, p3 being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) st f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) = <*p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(3 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) holds
L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = (LSeg (p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) \/ (LSeg (p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL3:17
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) )
for j, i being ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) st j : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) in dom (f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) | i : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V137() V138() V139() V140() V141() V142() ) Element of K19(NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) & j : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) in dom (f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) | i : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V137() V138() V139() V140() V141() V142() ) Element of K19(NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) holds
LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ,j : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = LSeg ((f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) | i : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,j : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL3:18
for f, h being ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) )
for j being ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) st j : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) in dom f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V137() V138() V139() V140() V141() V142() ) Element of K19(NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) & j : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) in dom f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V137() V138() V139() V140() V141() V142() ) Element of K19(NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) holds
LSeg ((f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ^ h : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,j : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ,j : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL3:19
for n being ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) )
for i being ( ( natural ) ( ordinal natural V22() real ext-real non negative ) Nat) holds LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ,i : ( ( natural ) ( ordinal natural V22() real ext-real non negative ) Nat) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL3:20
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) )
for i being ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) holds L~ (f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) | i : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL3:21
for r being ( ( real ) ( V22() real ext-real ) number )
for n being ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) )
for p1, p2 being ( ( ) ( V44(b2 : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) )
for u being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st p1 : ( ( ) ( V44(b2 : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid b2 : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & p2 : ( ( ) ( V44(b2 : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid b2 : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
LSeg (p1 : ( ( ) ( V44(b2 : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V44(b2 : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL b2 : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid b2 : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL3:22
for p1, p2, p being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) )
for r1, s1, r2, s2, r being ( ( real ) ( V22() real ext-real ) number )
for u being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) & p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) = |[r1 : ( ( real ) ( V22() real ext-real ) number ) ,s1 : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) = |[r2 : ( ( real ) ( V22() real ext-real ) number ) ,s2 : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) = |[r2 : ( ( real ) ( V22() real ext-real ) number ) ,s1 : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) & p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL3:23
for s, r1, r, s1 being ( ( real ) ( V22() real ext-real ) number )
for u being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st |[s : ( ( real ) ( V22() real ext-real ) number ) ,r1 : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & |[s : ( ( real ) ( V22() real ext-real ) number ) ,s1 : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
|[s : ( ( real ) ( V22() real ext-real ) number ) ,((r1 : ( ( real ) ( V22() real ext-real ) number ) + s1 : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) set ) / 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL3:24
for r1, s, r, s1 being ( ( real ) ( V22() real ext-real ) number )
for u being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st |[r1 : ( ( real ) ( V22() real ext-real ) number ) ,s : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & |[s1 : ( ( real ) ( V22() real ext-real ) number ) ,s : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
|[((r1 : ( ( real ) ( V22() real ext-real ) number ) + s1 : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) set ) / 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,s : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL3:25
for r1, r2, r, s1, s2 being ( ( real ) ( V22() real ext-real ) number )
for u being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st |[r1 : ( ( real ) ( V22() real ext-real ) number ) ,r2 : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & |[s1 : ( ( real ) ( V22() real ext-real ) number ) ,s2 : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & not |[r1 : ( ( real ) ( V22() real ext-real ) number ) ,s2 : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
|[s1 : ( ( real ) ( V22() real ext-real ) number ) ,r2 : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL3:26
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) )
for r being ( ( real ) ( V22() real ext-real ) number )
for u being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for m being ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) st not f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) /. 1 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & 1 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) & m : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) <= (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & ( for i being ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) <= (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & (LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ,i : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) )) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) <> {} : ( ( ) ( functional empty FinSequence-membered V137() V138() V139() V140() V141() V142() V143() ) set ) holds
m : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
not f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) /. m : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL3:27
for q, p2, p being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) st q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) <> p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) holds
((LSeg (p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,|[(p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) \/ (LSeg (|[(p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (LSeg (q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) ;

theorem :: TOPREAL3:28
for q, p2, p being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) st q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) <> p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) holds
((LSeg (p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,|[(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) \/ (LSeg (|[(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (LSeg (q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) ;

theorem :: TOPREAL3:29
for p, q being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) holds (LSeg (p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,|[(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (LSeg (|[(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {|[(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) ;

theorem :: TOPREAL3:30
for p, q being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) holds (LSeg (p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,|[(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (LSeg (|[(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {|[(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) ;

theorem :: TOPREAL3:31
for p, q being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) <> q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) holds
(LSeg (p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,|[(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(((p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) + (q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (LSeg (|[(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(((p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) + (q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {|[(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(((p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) + (q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) ;

theorem :: TOPREAL3:32
for p, q being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) <> q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) holds
(LSeg (p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,|[(((p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) + (q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (LSeg (|[(((p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) + (q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {|[(((p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) + (q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) ;

theorem :: TOPREAL3:33
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) )
for i being ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) st i : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) > 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) in dom f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V137() V138() V139() V140() V141() V142() ) Element of K19(NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) is being_S-Seq holds
f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) | i : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) is being_S-Seq ;

theorem :: TOPREAL3:34
for p, q being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) <> q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) <> q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) = <*p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,|[(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(3 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) holds
( f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) /. 1 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) = p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) = q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) is being_S-Seq ) ;

theorem :: TOPREAL3:35
for p, q being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) <> q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) <> q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) = <*p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,|[(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(3 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) holds
( f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) /. 1 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) = p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) = q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) is being_S-Seq ) ;

theorem :: TOPREAL3:36
for p, q being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) <> q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) = <*p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,|[(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(((p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) + (q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(3 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) holds
( f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) /. 1 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) = p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) = q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) is being_S-Seq ) ;

theorem :: TOPREAL3:37
for p, q being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) <> q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) = <*p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,|[(((p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) + (q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(3 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) holds
( f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) /. 1 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) = p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) = q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) is being_S-Seq ) ;

theorem :: TOPREAL3:38
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) )
for i being ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) st i : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) in dom f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V137() V138() V139() V140() V141() V142() ) Element of K19(NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) & i : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) in dom f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V137() V138() V139() V140() V141() V142() ) Element of K19(NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) holds
L~ (f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) | (i : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = (L~ (f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) | i : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) \/ (LSeg ((f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) /. i : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) /. (i : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL3:39
for p being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) st len f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) >= 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) & not p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) in L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
for n being ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) & n : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) holds
f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) /. n : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) <> p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TOPREAL3:40
for q, p being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) st q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) <> p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) & (LSeg (q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) holds
not p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) in L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPREAL3:41
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) )
for m being ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) st f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) is being_S-Seq & f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ,m : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & 1 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) & m : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) holds
m : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) = len f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: TOPREAL3:42
for p1, q, p being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) )
for r being ( ( real ) ( V22() real ext-real ) number )
for u being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st not p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & not p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( ( q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) <> p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) or ( q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) <> p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) ) & ( p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) or p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) holds
(LSeg (p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (LSeg (q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) ;

theorem :: TOPREAL3:43
for p1, p, q being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) )
for r being ( ( real ) ( V22() real ext-real ) number )
for u being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st not p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & |[(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & not |[(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) <> q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) <> q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) holds
((LSeg (p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,|[(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) \/ (LSeg (|[(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (LSeg (p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) ;

theorem :: TOPREAL3:44
for p1, p, q being ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) )
for r being ( ( real ) ( V22() real ext-real ) number )
for u being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st not p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & |[(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in Ball (u : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( ) Element of K19( the carrier of (Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & not |[(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in LSeg (p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) = p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) <> q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) & p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) <> q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) holds
((LSeg (p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,|[(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) \/ (LSeg (|[(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (LSeg (p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) ;

theorem :: TOPREAL3:45
for n being ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) FinSequence of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) st len f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) FinSequence of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) = n : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) holds
f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) FinSequence of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) in the carrier of (Euclid n : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict Reflexive discerning V86() triangle ) ( non empty strict Reflexive discerning V86() triangle ) MetrStruct ) : ( ( ) ( non empty ) set ) ;

theorem :: TOPREAL3:46
for n being ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) FinSequence of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) st len f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) FinSequence of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) = n : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) holds
f : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) -valued Function-like V37() FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) FinSequence of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) in the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V22() real ext-real non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ;