:: JGRAPH_7 semantic presentation

begin

theorem :: JGRAPH_7:1
for a, b, d being ( ( real ) ( V11() real ext-real ) number )
for p being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & p : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
p : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) in LSeg (|[a : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ]| : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,|[b : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ]| : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:2
for n being ( ( ) ( natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for P being ( ( ) ( ) Subset of )
for p1, p2 being ( ( ) ( V43(b1 : ( ( ) ( natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) st P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(b1 : ( ( ) ( natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(b1 : ( ( ) ( natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) holds
ex f being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st
( f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = P : ( ( ) ( ) Subset of ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(b1 : ( ( ) ( natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(b1 : ( ( ) ( natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: JGRAPH_7:3
for p1, p2 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for b, c, d being ( ( real ) ( V11() real ext-real ) number ) st p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < b : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) holds
LE p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) , rectangle ((p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:4
for p1, p2 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for b, c being ( ( real ) ( V11() real ext-real ) number ) st p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
LE p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) , rectangle ((p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,(p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:5
for p1, p2 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for c, d being ( ( real ) ( V11() real ext-real ) number ) st p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) holds
LE p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) , rectangle ((p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:6
for p1, p2 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for b, d being ( ( real ) ( V11() real ext-real ) number ) st p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
LE p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) , rectangle ((p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,(p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:7
for p1, p2 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
LE p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) , rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:8
for p1, p2 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) holds
LE p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) , rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:9
for p1, p2 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
LE p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) , rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:10
for p1, p2 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) holds
LE p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) , rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:11
for p1, p2 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
LE p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) , rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:12
for p1, p2 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
LE p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) , rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:13
for p1, p2 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) holds
LE p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) , rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:14
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:15
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:16
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:17
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:18
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:19
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:20
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:21
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:22
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:23
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:24
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:25
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:26
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:27
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:28
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) st p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <> p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <> p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle ((p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:29
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:30
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & d : ( ( real ) ( V11() real ext-real ) number ) >= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) >= c : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:31
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & d : ( ( real ) ( V11() real ext-real ) number ) >= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) >= c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:32
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:33
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:34
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:35
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:36
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:37
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:38
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:39
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:40
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & d : ( ( real ) ( V11() real ext-real ) number ) >= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) >= c : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:41
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & d : ( ( real ) ( V11() real ext-real ) number ) >= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) >= c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:42
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:43
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:44
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & d : ( ( real ) ( V11() real ext-real ) number ) >= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) >= c : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:45
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & d : ( ( real ) ( V11() real ext-real ) number ) >= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) >= c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:46
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & d : ( ( real ) ( V11() real ext-real ) number ) >= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) >= c : ( ( real ) ( V11() real ext-real ) number ) & b : ( ( real ) ( V11() real ext-real ) number ) >= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > a : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:47
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & b : ( ( real ) ( V11() real ext-real ) number ) >= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > a : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:48
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & b : ( ( real ) ( V11() real ext-real ) number ) >= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > a : ( ( real ) ( V11() real ext-real ) number ) holds
p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) are_in_this_order_on rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:49
for A, B, C, D being ( ( real ) ( V11() real ext-real ) number )
for h, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st A : ( ( real ) ( V11() real ext-real ) number ) > 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & C : ( ( real ) ( V11() real ext-real ) number ) > 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = AffineMap (A : ( ( real ) ( V11() real ext-real ) number ) ,B : ( ( real ) ( V11() real ext-real ) number ) ,C : ( ( real ) ( V11() real ext-real ) number ) ,D : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total continuous ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = AffineMap ((1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / A : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- (B : ( ( real ) ( V11() real ext-real ) number ) / A : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ,(1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / C : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- (D : ( ( real ) ( V11() real ext-real ) number ) / C : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ) : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total continuous ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
( g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) " : ( ( V19() Function-like ) ( V19() Function-like ) set ) & h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) " : ( ( V19() Function-like ) ( V19() Function-like ) set ) ) ;

theorem :: JGRAPH_7:50
for A, B, C, D being ( ( real ) ( V11() real ext-real ) number )
for h being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st A : ( ( real ) ( V11() real ext-real ) number ) > 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & C : ( ( real ) ( V11() real ext-real ) number ) > 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = AffineMap (A : ( ( real ) ( V11() real ext-real ) number ) ,B : ( ( real ) ( V11() real ext-real ) number ) ,C : ( ( real ) ( V11() real ext-real ) number ) ,D : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total continuous ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
( h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is being_homeomorphism & ( for p1, p2 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) st p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) holds
(h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < (h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ) ) ;

theorem :: JGRAPH_7:51
for A, B, C, D being ( ( real ) ( V11() real ext-real ) number )
for h being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st A : ( ( real ) ( V11() real ext-real ) number ) > 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & C : ( ( real ) ( V11() real ext-real ) number ) > 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = AffineMap (A : ( ( real ) ( V11() real ext-real ) number ) ,B : ( ( real ) ( V11() real ext-real ) number ) ,C : ( ( real ) ( V11() real ext-real ) number ) ,D : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total continuous ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
( h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is being_homeomorphism & ( for p1, p2 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) st p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) holds
(h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < (h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ) ) ;

theorem :: JGRAPH_7:52
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for h being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = AffineMap ((2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((b : ( ( real ) ( V11() real ext-real ) number ) + a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ,(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((d : ( ( real ) ( V11() real ext-real ) number ) + c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ) : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total continuous ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng (h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle ((- 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(- 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:53
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for h being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = AffineMap ((2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((b : ( ( real ) ( V11() real ext-real ) number ) + a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ,(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((d : ( ( real ) ( V11() real ext-real ) number ) + c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ) : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total continuous ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one holds
( h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is continuous & h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is one-to-one ) ;

theorem :: JGRAPH_7:54
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for h being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for O being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = AffineMap ((2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((b : ( ( real ) ( V11() real ext-real ) number ) + a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ,(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((d : ( ( real ) ( V11() real ext-real ) number ) + c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ) : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total continuous ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) holds
((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = - 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ;

theorem :: JGRAPH_7:55
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for h being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for I being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = AffineMap ((2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((b : ( ( real ) ( V11() real ext-real ) number ) + a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ,(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((d : ( ( real ) ( V11() real ext-real ) number ) + c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ) : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total continuous ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) holds
((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JGRAPH_7:56
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for h being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for I being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = AffineMap ((2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((b : ( ( real ) ( V11() real ext-real ) number ) + a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ,(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((d : ( ( real ) ( V11() real ext-real ) number ) + c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ) : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total continuous ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) holds
((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JGRAPH_7:57
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for h being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for I being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = AffineMap ((2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((b : ( ( real ) ( V11() real ext-real ) number ) + a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ,(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((d : ( ( real ) ( V11() real ext-real ) number ) + c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ) : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total continuous ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) holds
((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = - 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ;

theorem :: JGRAPH_7:58
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for h being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for O, I being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = AffineMap ((2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((b : ( ( real ) ( V11() real ext-real ) number ) + a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ,(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((d : ( ( real ) ( V11() real ext-real ) number ) + c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ) : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total continuous ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & c : ( ( real ) ( V11() real ext-real ) number ) <= (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) holds
( - 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: JGRAPH_7:59
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for h being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for O, I being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = AffineMap ((2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((b : ( ( real ) ( V11() real ext-real ) number ) + a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ,(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((d : ( ( real ) ( V11() real ext-real ) number ) + c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ) : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total continuous ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & c : ( ( real ) ( V11() real ext-real ) number ) <= (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
( - 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & - 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: JGRAPH_7:60
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for h being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for O, I being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = AffineMap ((2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((b : ( ( real ) ( V11() real ext-real ) number ) + a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ,(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((d : ( ( real ) ( V11() real ext-real ) number ) + c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ) : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total continuous ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & c : ( ( real ) ( V11() real ext-real ) number ) <= (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) holds
( - 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & - 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: JGRAPH_7:61
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for h being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for O, I being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = AffineMap ((2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((b : ( ( real ) ( V11() real ext-real ) number ) + a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ,(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((d : ( ( real ) ( V11() real ext-real ) number ) + c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ) : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total continuous ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & c : ( ( real ) ( V11() real ext-real ) number ) <= (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
( - 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & - 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: JGRAPH_7:62
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for h being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for O, I being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = AffineMap ((2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((b : ( ( real ) ( V11() real ext-real ) number ) + a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ,(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((d : ( ( real ) ( V11() real ext-real ) number ) + c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ) : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total continuous ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & a : ( ( real ) ( V11() real ext-real ) number ) <= (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
( - 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: JGRAPH_7:63
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for h being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for O, I being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = AffineMap ((2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((b : ( ( real ) ( V11() real ext-real ) number ) + a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ,(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((d : ( ( real ) ( V11() real ext-real ) number ) + c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ) : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total continuous ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & a : ( ( real ) ( V11() real ext-real ) number ) <= (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) holds
( - 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & - 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: JGRAPH_7:64
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for h being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for O, I being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = AffineMap ((2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((b : ( ( real ) ( V11() real ext-real ) number ) + a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ,(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((d : ( ( real ) ( V11() real ext-real ) number ) + c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ) : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total continuous ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & a : ( ( real ) ( V11() real ext-real ) number ) <= (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
( - 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & - 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: JGRAPH_7:65
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for h being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for O, I being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = AffineMap ((2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((b : ( ( real ) ( V11() real ext-real ) number ) + a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ,(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((d : ( ( real ) ( V11() real ext-real ) number ) + c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ) : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total continuous ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & d : ( ( real ) ( V11() real ext-real ) number ) >= (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) >= c : ( ( real ) ( V11() real ext-real ) number ) holds
( 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) >= ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) >= - 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ) ;

theorem :: JGRAPH_7:66
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for h being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for O, I being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = AffineMap ((2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((b : ( ( real ) ( V11() real ext-real ) number ) + a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ,(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((d : ( ( real ) ( V11() real ext-real ) number ) + c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ) : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total continuous ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & c : ( ( real ) ( V11() real ext-real ) number ) <= (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
( - 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & - 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: JGRAPH_7:67
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for h being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for O, I being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = AffineMap ((2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((b : ( ( real ) ( V11() real ext-real ) number ) + a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (b : ( ( real ) ( V11() real ext-real ) number ) - a : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ,(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) ,(- ((d : ( ( real ) ( V11() real ext-real ) number ) + c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (d : ( ( real ) ( V11() real ext-real ) number ) - c : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) ) : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total continuous ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & a : ( ( real ) ( V11() real ext-real ) number ) < (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & (f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) holds
( - 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & ((h : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of K6(K7( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: JGRAPH_7:68
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:69
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:70
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:71
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:72
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:73
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:74
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:75
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:76
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:77
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:78
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:79
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:80
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:81
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:82
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:83
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:84
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:85
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:86
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:87
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:88
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:89
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:90
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:91
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:92
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:93
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:94
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:95
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:96
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:97
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:98
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:99
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:100
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:101
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:102
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:103
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:104
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:105
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:106
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:107
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = a : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:108
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:109
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:110
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:111
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:112
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:113
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:114
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:115
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:116
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:117
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:118
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:119
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:120
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & d : ( ( real ) ( V11() real ext-real ) number ) >= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) >= c : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:121
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & d : ( ( real ) ( V11() real ext-real ) number ) >= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) >= c : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:122
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & d : ( ( real ) ( V11() real ext-real ) number ) >= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) >= c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:123
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & d : ( ( real ) ( V11() real ext-real ) number ) >= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) >= c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:124
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:125
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:126
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:127
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = d : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) < p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:128
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & d : ( ( real ) ( V11() real ext-real ) number ) >= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) >= c : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:129
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & d : ( ( real ) ( V11() real ext-real ) number ) >= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) >= c : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:130
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & d : ( ( real ) ( V11() real ext-real ) number ) >= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) >= c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:131
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & d : ( ( real ) ( V11() real ext-real ) number ) >= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) >= c : ( ( real ) ( V11() real ext-real ) number ) & a : ( ( real ) ( V11() real ext-real ) number ) < p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= b : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:132
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & d : ( ( real ) ( V11() real ext-real ) number ) >= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) >= c : ( ( real ) ( V11() real ext-real ) number ) & b : ( ( real ) ( V11() real ext-real ) number ) >= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > a : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:133
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & d : ( ( real ) ( V11() real ext-real ) number ) >= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) >= c : ( ( real ) ( V11() real ext-real ) number ) & b : ( ( real ) ( V11() real ext-real ) number ) >= p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > a : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:134
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & b : ( ( real ) ( V11() real ext-real ) number ) >= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > a : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:135
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = b : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) <= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) <= d : ( ( real ) ( V11() real ext-real ) number ) & b : ( ( real ) ( V11() real ext-real ) number ) >= p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > a : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;

theorem :: JGRAPH_7:136
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for f, g being ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & b : ( ( real ) ( V11() real ext-real ) number ) >= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > a : ( ( real ) ( V11() real ext-real ) number ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( empty natural V11() real ext-real V112() V113() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like quasi_total ) ( non empty V19() V22( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like V29( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K255() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_7:137
for p1, p2, p3, p4 being ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( real ) ( V11() real ext-real ) number )
for P, Q being ( ( ) ( ) Subset of ) st a : ( ( real ) ( V11() real ext-real ) number ) < b : ( ( real ) ( V11() real ext-real ) number ) & c : ( ( real ) ( V11() real ext-real ) number ) < d : ( ( real ) ( V11() real ext-real ) number ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) = c : ( ( real ) ( V11() real ext-real ) number ) & b : ( ( real ) ( V11() real ext-real ) number ) >= p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) & p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) > a : ( ( real ) ( V11() real ext-real ) number ) & P : ( ( ) ( ) Subset of ) is_an_arc_of p1 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p3 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & Q : ( ( ) ( ) Subset of ) is_an_arc_of p2 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) ,p4 : ( ( ) ( V43(2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) V109() V152() ) Point of ( ( ) ( non empty ) set ) ) & P : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Q : ( ( ) ( ) Subset of ) c= closed_inside_of_rectangle (a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) ,c : ( ( real ) ( V11() real ext-real ) number ) ,d : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty natural V11() real ext-real positive V112() V113() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V126() V172() V173() V174() V175() V176() V177() V178() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
P : ( ( ) ( ) Subset of ) meets Q : ( ( ) ( ) Subset of ) ;