:: JORDAN5C semantic presentation

begin

theorem :: JORDAN5C:1
for P, Q being ( ( ) ( functional ) Subset of )
for p1, p2, q1 being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) )
for f being ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) )
for s1 being ( ( ) ( V11() V12() ext-real ) Real) st q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in P : ( ( ) ( functional ) Subset of ) & f : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . s1 : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( ) set ) = q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & f : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) is being_homeomorphism & f : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative Function-like functional V79() V80() V142() V143() V144() V145() V146() V147() V148() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & f : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative Function-like functional V79() V80() V142() V143() V144() V145() V146() V147() V148() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= s1 : ( ( ) ( V11() V12() ext-real ) Real) & s1 : ( ( ) ( V11() V12() ext-real ) Real) <= 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for t being ( ( ) ( V11() V12() ext-real ) Real) st 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative Function-like functional V79() V80() V142() V143() V144() V145() V146() V147() V148() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= t : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) & t : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) < s1 : ( ( ) ( V11() V12() ext-real ) Real) holds
not f : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . t : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) : ( ( ) ( ) set ) in Q : ( ( ) ( functional ) Subset of ) ) holds
for g being ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) )
for s2 being ( ( ) ( V11() V12() ext-real ) Real) st g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) is being_homeomorphism & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative Function-like functional V79() V80() V142() V143() V144() V145() V146() V147() V148() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . s2 : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( ) set ) = q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative Function-like functional V79() V80() V142() V143() V144() V145() V146() V147() V148() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= s2 : ( ( ) ( V11() V12() ext-real ) Real) & s2 : ( ( ) ( V11() V12() ext-real ) Real) <= 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for t being ( ( ) ( V11() V12() ext-real ) Real) st 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative Function-like functional V79() V80() V142() V143() V144() V145() V146() V147() V148() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= t : ( ( ) ( V11() V12() ext-real ) Real) & t : ( ( ) ( V11() V12() ext-real ) Real) < s2 : ( ( ) ( V11() V12() ext-real ) Real) holds
not g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . t : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( ) set ) in Q : ( ( ) ( functional ) Subset of ) ;

definition
let P, Q be ( ( ) ( functional ) Subset of ) ;
let p1, p2 be ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ;
assume that
( P : ( ( ) ( functional ) Subset of ) meets Q : ( ( ) ( functional ) Subset of ) & P : ( ( ) ( functional ) Subset of ) /\ Q : ( ( ) ( functional ) Subset of ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) is closed ) and
P : ( ( ) ( functional ) Subset of ) is_an_arc_of p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ;
func First_Point (P,p1,p2,Q) -> ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) means :: JORDAN5C:def 1
( it : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) in P : ( ( ) ( ) RLTopStruct ) /\ Q : ( ( ) ( ) Element of P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) & ( for g being ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | P : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) )
for s2 being ( ( ) ( V11() V12() ext-real ) Real) st g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | P : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) is being_homeomorphism & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | P : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative Function-like functional V79() V80() V142() V143() V144() V145() V146() V147() V148() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( Function-like quasi_total ) ( Relation-like K7(P : ( ( ) ( ) RLTopStruct ) ,P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined P : ( ( ) ( ) RLTopStruct ) -valued Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) RLTopStruct ) ,P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | P : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ,P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined P : ( ( ) ( ) RLTopStruct ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ,P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | P : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . s2 : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( ) set ) = it : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative Function-like functional V79() V80() V142() V143() V144() V145() V146() V147() V148() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= s2 : ( ( ) ( V11() V12() ext-real ) Real) & s2 : ( ( ) ( V11() V12() ext-real ) Real) <= 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for t being ( ( ) ( V11() V12() ext-real ) Real) st 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative Function-like functional V79() V80() V142() V143() V144() V145() V146() V147() V148() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= t : ( ( ) ( V11() V12() ext-real ) Real) & t : ( ( ) ( V11() V12() ext-real ) Real) < s2 : ( ( ) ( V11() V12() ext-real ) Real) holds
not g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | P : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . t : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( ) set ) in Q : ( ( ) ( ) Element of P : ( ( ) ( ) RLTopStruct ) ) ) );
end;

theorem :: JORDAN5C:2
for P, Q being ( ( ) ( functional ) Subset of )
for p, p1, p2 being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) st p : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in P : ( ( ) ( functional ) Subset of ) & P : ( ( ) ( functional ) Subset of ) is_an_arc_of p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & Q : ( ( ) ( functional ) Subset of ) = {p : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty functional ) set ) holds
First_Point (P : ( ( ) ( functional ) Subset of ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) = p : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN5C:3
for P, Q being ( ( ) ( functional ) Subset of )
for p1, p2 being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) st p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in Q : ( ( ) ( functional ) Subset of ) & P : ( ( ) ( functional ) Subset of ) /\ Q : ( ( ) ( functional ) Subset of ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) is closed & P : ( ( ) ( functional ) Subset of ) is_an_arc_of p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) holds
First_Point (P : ( ( ) ( functional ) Subset of ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) = p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN5C:4
for P, Q being ( ( ) ( functional ) Subset of )
for p1, p2, q1 being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) )
for f being ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) )
for s1 being ( ( ) ( V11() V12() ext-real ) Real) st q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in P : ( ( ) ( functional ) Subset of ) & f : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . s1 : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( ) set ) = q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & f : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) is being_homeomorphism & f : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative Function-like functional V79() V80() V142() V143() V144() V145() V146() V147() V148() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & f : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative Function-like functional V79() V80() V142() V143() V144() V145() V146() V147() V148() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= s1 : ( ( ) ( V11() V12() ext-real ) Real) & s1 : ( ( ) ( V11() V12() ext-real ) Real) <= 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for t being ( ( ) ( V11() V12() ext-real ) Real) st 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) >= t : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) & t : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) > s1 : ( ( ) ( V11() V12() ext-real ) Real) holds
not f : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . t : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) : ( ( ) ( ) set ) in Q : ( ( ) ( functional ) Subset of ) ) holds
for g being ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) )
for s2 being ( ( ) ( V11() V12() ext-real ) Real) st g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) is being_homeomorphism & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative Function-like functional V79() V80() V142() V143() V144() V145() V146() V147() V148() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . s2 : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( ) set ) = q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative Function-like functional V79() V80() V142() V143() V144() V145() V146() V147() V148() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= s2 : ( ( ) ( V11() V12() ext-real ) Real) & s2 : ( ( ) ( V11() V12() ext-real ) Real) <= 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for t being ( ( ) ( V11() V12() ext-real ) Real) st 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) >= t : ( ( ) ( V11() V12() ext-real ) Real) & t : ( ( ) ( V11() V12() ext-real ) Real) > s2 : ( ( ) ( V11() V12() ext-real ) Real) holds
not g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . t : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( ) set ) in Q : ( ( ) ( functional ) Subset of ) ;

definition
let P, Q be ( ( ) ( functional ) Subset of ) ;
let p1, p2 be ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ;
assume that
( P : ( ( ) ( functional ) Subset of ) meets Q : ( ( ) ( functional ) Subset of ) & P : ( ( ) ( functional ) Subset of ) /\ Q : ( ( ) ( functional ) Subset of ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) is closed ) and
P : ( ( ) ( functional ) Subset of ) is_an_arc_of p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ;
func Last_Point (P,p1,p2,Q) -> ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) means :: JORDAN5C:def 2
( it : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) in P : ( ( ) ( ) RLTopStruct ) /\ Q : ( ( ) ( ) Element of P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) & ( for g being ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | P : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) )
for s2 being ( ( ) ( V11() V12() ext-real ) Real) st g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | P : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) is being_homeomorphism & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | P : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative Function-like functional V79() V80() V142() V143() V144() V145() V146() V147() V148() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( Function-like quasi_total ) ( Relation-like K7(P : ( ( ) ( ) RLTopStruct ) ,P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined P : ( ( ) ( ) RLTopStruct ) -valued Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) RLTopStruct ) ,P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | P : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ,P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined P : ( ( ) ( ) RLTopStruct ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ,P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | P : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . s2 : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( ) set ) = it : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative Function-like functional V79() V80() V142() V143() V144() V145() V146() V147() V148() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= s2 : ( ( ) ( V11() V12() ext-real ) Real) & s2 : ( ( ) ( V11() V12() ext-real ) Real) <= 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for t being ( ( ) ( V11() V12() ext-real ) Real) st 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) >= t : ( ( ) ( V11() V12() ext-real ) Real) & t : ( ( ) ( V11() V12() ext-real ) Real) > s2 : ( ( ) ( V11() V12() ext-real ) Real) holds
not g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | P : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . t : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( ) set ) in Q : ( ( ) ( ) Element of P : ( ( ) ( ) RLTopStruct ) ) ) );
end;

theorem :: JORDAN5C:5
for P, Q being ( ( ) ( functional ) Subset of )
for p, p1, p2 being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) st p : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in P : ( ( ) ( functional ) Subset of ) & P : ( ( ) ( functional ) Subset of ) is_an_arc_of p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & Q : ( ( ) ( functional ) Subset of ) = {p : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty functional ) set ) holds
Last_Point (P : ( ( ) ( functional ) Subset of ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) = p : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN5C:6
for P, Q being ( ( ) ( functional ) Subset of )
for p1, p2 being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) st p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in Q : ( ( ) ( functional ) Subset of ) & P : ( ( ) ( functional ) Subset of ) /\ Q : ( ( ) ( functional ) Subset of ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) is closed & P : ( ( ) ( functional ) Subset of ) is_an_arc_of p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) holds
Last_Point (P : ( ( ) ( functional ) Subset of ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) = p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN5C:7
for P, Q being ( ( ) ( functional ) Subset of )
for p1, p2 being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( functional ) Subset of ) c= Q : ( ( ) ( functional ) Subset of ) & P : ( ( ) ( functional ) Subset of ) is closed & P : ( ( ) ( functional ) Subset of ) is_an_arc_of p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) holds
( First_Point (P : ( ( ) ( functional ) Subset of ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) = p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & Last_Point (P : ( ( ) ( functional ) Subset of ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) = p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ) ;

begin

definition
let P be ( ( ) ( functional ) Subset of ) ;
let p1, p2, q1, q2 be ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ;
pred LE q1,q2,P,p1,p2 means :: JORDAN5C:def 3
( q1 : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ,P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined P : ( ( ) ( ) RLTopStruct ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ,P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) in P : ( ( ) ( ) RLTopStruct ) & q2 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) in P : ( ( ) ( ) RLTopStruct ) & ( for g being ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | P : ( ( ) ( ) RLTopStruct ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) )
for s1, s2 being ( ( ) ( V11() V12() ext-real ) Real) st g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | P : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) is being_homeomorphism & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | P : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative Function-like functional V79() V80() V142() V143() V144() V145() V146() V147() V148() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( ) Element of P : ( ( ) ( ) RLTopStruct ) ) & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | P : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( Function-like quasi_total ) ( Relation-like K7(P : ( ( ) ( ) RLTopStruct ) ,P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined P : ( ( ) ( ) RLTopStruct ) -valued Function-like quasi_total ) Element of K6(K7(K7(P : ( ( ) ( ) RLTopStruct ) ,P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | P : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . s1 : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( ) set ) = q1 : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ,P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined P : ( ( ) ( ) RLTopStruct ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ,P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ,P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative Function-like functional V79() V80() V142() V143() V144() V145() V146() V147() V148() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= s1 : ( ( ) ( V11() V12() ext-real ) Real) & s1 : ( ( ) ( V11() V12() ext-real ) Real) <= 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | P : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . s2 : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( ) set ) = q2 : ( ( ) ( ) Element of K6(K6(P : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative Function-like functional V79() V80() V142() V143() V144() V145() V146() V147() V148() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= s2 : ( ( ) ( V11() V12() ext-real ) Real) & s2 : ( ( ) ( V11() V12() ext-real ) Real) <= 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) holds
s1 : ( ( ) ( V11() V12() ext-real ) Real) <= s2 : ( ( ) ( V11() V12() ext-real ) Real) ) );
end;

theorem :: JORDAN5C:8
for P being ( ( ) ( functional ) Subset of )
for p1, p2, q1, q2 being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) )
for g being ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) )
for s1, s2 being ( ( ) ( V11() V12() ext-real ) Real) st P : ( ( ) ( functional ) Subset of ) is_an_arc_of p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) is being_homeomorphism & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative Function-like functional V79() V80() V142() V143() V144() V145() V146() V147() V148() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . s1 : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( ) set ) = q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative Function-like functional V79() V80() V142() V143() V144() V145() V146() V147() V148() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= s1 : ( ( ) ( V11() V12() ext-real ) Real) & s1 : ( ( ) ( V11() V12() ext-real ) Real) <= 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of R^1 : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) | b1 : ( ( ) ( functional ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) . s2 : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( ) set ) = q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & 0 : ( ( ) ( empty V10() V11() V12() ext-real non positive non negative Function-like functional V79() V80() V142() V143() V144() V145() V146() V147() V148() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= s2 : ( ( ) ( V11() V12() ext-real ) Real) & s2 : ( ( ) ( V11() V12() ext-real ) Real) <= 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & s1 : ( ( ) ( V11() V12() ext-real ) Real) <= s2 : ( ( ) ( V11() V12() ext-real ) Real) holds
LE q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( functional ) Subset of ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN5C:9
for P being ( ( ) ( functional ) Subset of )
for p1, p2, q1 being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) st q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in P : ( ( ) ( functional ) Subset of ) holds
LE q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( functional ) Subset of ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN5C:10
for P being ( ( ) ( functional ) Subset of )
for p1, p2, q1 being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( functional ) Subset of ) is_an_arc_of p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in P : ( ( ) ( functional ) Subset of ) holds
( LE p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( functional ) Subset of ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & LE q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( functional ) Subset of ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ) ;

theorem :: JORDAN5C:11
for P being ( ( ) ( functional ) Subset of )
for p1, p2 being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( functional ) Subset of ) is_an_arc_of p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) holds
LE p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( functional ) Subset of ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN5C:12
for P being ( ( ) ( functional ) Subset of )
for p1, p2, q1, q2 being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( functional ) Subset of ) is_an_arc_of p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & LE q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( functional ) Subset of ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & LE q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( functional ) Subset of ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) holds
q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) = q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN5C:13
for P being ( ( ) ( functional ) Subset of )
for p1, p2, q1, q2, q3 being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) st LE q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( functional ) Subset of ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & LE q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,q3 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( functional ) Subset of ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) holds
LE q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,q3 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( functional ) Subset of ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN5C:14
for P being ( ( ) ( functional ) Subset of )
for p1, p2, q1, q2 being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( functional ) Subset of ) is_an_arc_of p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in P : ( ( ) ( functional ) Subset of ) & q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in P : ( ( ) ( functional ) Subset of ) & q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) <> q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & not ( LE q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( functional ) Subset of ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & not LE q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( functional ) Subset of ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ) holds
( LE q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( functional ) Subset of ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & not LE q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,P : ( ( ) ( functional ) Subset of ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ) ;

begin

theorem :: JORDAN5C:15
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) )
for Q being ( ( ) ( functional ) Subset of )
for q being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) st f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) is being_S-Seq & (L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) /\ Q : ( ( ) ( functional ) Subset of ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) is closed & q : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) & q : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in Q : ( ( ) ( functional ) Subset of ) holds
LE First_Point ((L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,q : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) , L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN5C:16
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) )
for Q being ( ( ) ( functional ) Subset of )
for q being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) st f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) is being_S-Seq & (L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) /\ Q : ( ( ) ( functional ) Subset of ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) is closed & q : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) & q : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in Q : ( ( ) ( functional ) Subset of ) holds
LE q : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) , Last_Point ((L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) , L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN5C:17
for q1, q2, p1, p2 being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) st p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) <> p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & LE q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) , LSeg (p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( functional closed compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) holds
LE q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN5C:18
for P, Q being ( ( ) ( functional ) Subset of )
for p1, p2 being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) st P : ( ( ) ( functional ) Subset of ) is_an_arc_of p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & P : ( ( ) ( functional ) Subset of ) meets Q : ( ( ) ( functional ) Subset of ) & P : ( ( ) ( functional ) Subset of ) /\ Q : ( ( ) ( functional ) Subset of ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) is closed holds
( First_Point (P : ( ( ) ( functional ) Subset of ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) = Last_Point (P : ( ( ) ( functional ) Subset of ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) & Last_Point (P : ( ( ) ( functional ) Subset of ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) = First_Point (P : ( ( ) ( functional ) Subset of ) ,p2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,p1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ) ;

theorem :: JORDAN5C:19
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) )
for Q being ( ( ) ( functional ) Subset of )
for i being ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) st L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) meets Q : ( ( ) ( functional ) Subset of ) & Q : ( ( ) ( functional ) Subset of ) is closed & f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) is being_S-Seq & 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & First_Point ((L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ,i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) holds
First_Point ((L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) = First_Point ((LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ,i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN5C:20
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) )
for Q being ( ( ) ( functional ) Subset of )
for i being ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) st L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) meets Q : ( ( ) ( functional ) Subset of ) & Q : ( ( ) ( functional ) Subset of ) is closed & f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) is being_S-Seq & 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & Last_Point ((L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ,i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) holds
Last_Point ((L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) = Last_Point ((LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ,i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN5C:21
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) )
for i being ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) is being_S-Seq & First_Point ((L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ,i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ,i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) holds
First_Point ((L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ,i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) = f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN5C:22
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) )
for i being ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) is being_S-Seq & Last_Point ((L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ,i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ,i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) holds
Last_Point ((L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ,i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) = f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN5C:23
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) )
for i being ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) st f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) is being_S-Seq & 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) holds
LE f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) , L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN5C:24
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) )
for i, j being ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) st f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) is being_S-Seq & 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) holds
LE f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. j : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) , L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN5C:25
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) )
for q being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) )
for i being ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) st f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) is being_S-Seq & 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & q : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ,i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) holds
LE f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,q : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) , L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN5C:26
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) )
for q being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) )
for i being ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) st f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) is being_S-Seq & 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & q : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ,i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) holds
LE q : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) , L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN5C:27
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) )
for Q being ( ( ) ( functional ) Subset of )
for q being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) )
for i, j being ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) st L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) meets Q : ( ( ) ( functional ) Subset of ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) is being_S-Seq & Q : ( ( ) ( functional ) Subset of ) is closed & First_Point ((L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ,i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) & 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & q : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ,j : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) & 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & q : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in Q : ( ( ) ( functional ) Subset of ) & First_Point ((L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) <> q : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) holds
( i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & ( i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) = j : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) implies LE First_Point ((L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,q : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ) ) ;

theorem :: JORDAN5C:28
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) )
for Q being ( ( ) ( functional ) Subset of )
for q being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) )
for i, j being ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) st L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) meets Q : ( ( ) ( functional ) Subset of ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) is being_S-Seq & Q : ( ( ) ( functional ) Subset of ) is closed & Last_Point ((L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ,i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) & 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & q : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ,j : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) & 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & q : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in Q : ( ( ) ( functional ) Subset of ) & Last_Point ((L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) <> q : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) holds
( i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) >= j : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & ( i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) = j : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) implies LE q : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) , Last_Point ((L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,(f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,Q : ( ( ) ( functional ) Subset of ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ) ) ;

theorem :: JORDAN5C:29
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) )
for q1, q2 being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) )
for i being ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) st q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ,i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) & q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ,i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) is being_S-Seq & 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & LE q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) , L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) holds
LE q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) , LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ,i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;

theorem :: JORDAN5C:30
for f being ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) )
for q1, q2 being ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) st q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) & q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) is being_S-Seq & q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) <> q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) holds
( LE q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) , L~ f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) iff for i, j being ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) st q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ,i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) & q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) in LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) ,j : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( ) set ) ) & 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) & ( i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) = j : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) implies LE q1 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,q2 : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Point of ( ( ) ( non empty functional ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ,f : ( ( ) ( Relation-like NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty functional ) set ) ) /. (i : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V10() V11() V12() ext-real V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V43(2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V134() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V79() V80() V142() V143() V144() V145() V146() V147() ) Element of NAT : ( ( ) ( V142() V143() V144() V145() V146() V147() V148() ) Element of K6(REAL : ( ( ) ( V142() V143() V144() V148() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V108() V154() V155() V156() V157() V158() V159() V160() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ) ) ) ;